A Theory of Optimal Random Crackdowns
October 2008
Abstract: This paper develops an incentives-based theory of policing
that can explain the phenomenon of random “crackdowns,” which are intermittent periods of high interdiction/surveillance. We show that, when police
minimize the crime rate, random crackdowns can emerge as part of an optimal policing strategy. We develop several variations of the basic policing
model that would apply in different monitoring situations, such as speeding,
drug interdiction, or screening to deter terrorism. For a variety of police
objective functions, random crackdowns can be part of the optimal monitoring strategy. We demonstrate support for implications of the crackdown
theory using traffic data gathered by the Belgian Police Department and use
the model to estimate the deterrence effect of additional resources spent on
speeding interdiction.
2
1
Introduction
Police often engage in “crackdowns” on crime, which are intermittent periods of high
intensity policing. This paper develops a theoretical framework for modeling police monitoring behavior and individuals’ decisions to engage in crime. Within this framework,
we show that there are situations where it will be optimal for a crime-minimizing police
agency to engage in random crackdowns. When they occur, crackdowns also provide a
way of estimating the deterrence effect of policing. We illustrate the application of the
model in analyzing a speed deterrence program used by police in Belgium. In particular,
we estimate the deterrence effect of additional resources spent on ticketing speeders and
assess whether the current level of deterrence is socially optimal.
Two features characterize our notion of random crackdowns. First, they are arbitrary,
in the sense that they subject certain groups (identified by presence in a particular
time or place, or by other observable characteristic) that are not notably different from
other groups in criminal propensities, to higher intensity police monitoring. Second,
they are publicized, i.e., those who are subjected to crackdowns are informed about
them before they engage in criminal activity.1 Crackdowns are employed in a number
of policing situations. Some examples include drunk driving interdiction accomplished
using sobriety checkpoints, crackdowns on speeding achieved through announced greater
police presence on certain highways, or crackdowns on drug trafficking aimed at particular
neighborhoods.2
Criminologists sometimes rationalize the use of crackdowns by appealing to psychological theories according to which the impression created by the temporary show of
force (the crackdown) is a psychological “bluff” that leads potential criminals to overestimate the risk of detection during non-crackdown periods.3 This view relies on the
1
Our definition of crackdown is different from the conventional use of the term in the literature
on policing (see e.g. Di Tella and Schargrodsky 2002, 2004) because we require that crackdowns be
arbitrary. We will return to this point when we discuss the related literature.
2
For example, operation “safe streets” in Philadelphia, which puts heavy law enforcement on particular city blocks, received extensive media coverage. Other examples of crackdowns include the NHTSA
campaign “You Drink & Drive. You Lose” which instituted highly visible enforcement against drunk
driving. Another example is “Checkpoint Tennessee,” Tennessee’s statewide sobriety checkpoint program.
3
Sherman (1990), Ross (1984). Sherman, p. 11, recommends that crackdowns be highly publicized
3
potential criminals’ expectations being systematically wrong and therefore is inconsistent
with rationality.
In this paper, we take a different approach and develop a model in
which potential criminals are not fooled about the odds of detection and yet the crimeminimizing police finds it optimal to employ crackdowns. In our model, crackdowns arise
as a rational (indeed, optimal) response of both police and citizens’ incentives. They also
arise under a variety of ways of specifying police objectives, for example, if the police
minimize total crime, or undetected crime, or if they solve the social planner problem of
trading off the costs and benefits of crime.
We next illustrate the main idea behind this result through a simple example where
the police minimize crime subject to a resource constraint.
Example Consider a population of 100 citizens, half of whom would never commit
a crime, and half of whom would commit a crime unless they are certain that they will
be caught. A citizen’s propensity to commit a crime is unobservable to the police. The
police resources are such that they can only check 50 citizens. Suppose that the police
check citizens at random (note that all citizens look the same to police), so that each
citizen has a probability 1/2 of being checked. Then, only the high-propensity citizens will
commit a crime, giving rise to a crime rate of 1/2. Suppose now that half of the citizens
have blue eyes, half have brown eyes and that eye color is known to be independent of the
propensity to commit a crime. Nevertheless, suppose that police crack down on browneyed citizens and check them all, and completely ignore the blue-eyed citizens. Then no
brown-eyed ever commits a crime because they are sure that they would be caught, and
only those blue-eyed citizens commit a crime who have high criminal propensity. Thus,
the crime rate with a crackdown on brown-eyed persons is 1/4, which is lower than the
crime rate of 1/2 obtained without crackdowns.
This thought experiment shows that crackdowns can reduce crime by introducing
disparate treatment within a population of observably identical individuals. We have
not proved that the specific way in which citizens are divided and policed (blue-eyed v.
brown-eyed) is the optimal one for reducing crime, though this is indeed the case. We
show later in the paper that given any distribution (continuous or discrete, unimodal or
and be followed by secret “backdowns," and warns of the risk of exhausting the bluff through overuse.
4
multi-modal) of the propensity to commit a crime, the crime-minimizing policing scheme
involves dividing the population into no more than two groups, not necessarily of equal
size. The example also highlights an important maintained assumption of our theory:
for crackdowns to be effective, it is important that criminals cannot easily arbitrage
between crackdown and non-crackdown groups. In the example, citizens are assumed to
be unable to change their eye color.
Let us now return to the example to consider how crackdowns make it possible to
estimate the deterrence effect of policing.
Using crackdowns to identify the deterrence effect of policing Consider
now an increase in police manpower to 51 checks. How does the optimal policing scheme
change? It can be shown that the optimal policing scheme involves moving one person
from the non-crackdown group to the crackdown group. That is, police would pick a blueeyed citizen, force him to wear brown contact lenses, and then check with probability 1 all
those who appear to have brown eyes. The remaining citizens (with blue eyes) are never
checked. We can calculate the expected decrease in the crime rate that follows from an
increase in manpower of 1 check: it is the decrease in crime that obtains from moving a
random citizen from the group that is not cracked down upon to the group that is cracked
down upon. Because the average crime rate in both groups is observed, we can readily
compute the expected decrease in the crime rate – in this case, the expected crime rate
goes from 25 percent to 24.5 percent.
In this paper, we develop a model of policing in which a police chief is given an incentive to reduce crime and a certain amount of resources. Under a variety of assumptions
about police goals and constraints, the optimal monitoring strategy can take the form
of random crackdowns. As in the above example, our analysis provides a methodology
for estimating the deterrence effect of policing.
We apply our policing model to analyze the effectiveness of resources spent on speeding interdiction. Although the decision to speed is rarely studied by economists,4 it has
great economic relevance, both in the U.S. and worldwide. According to data from the
4
With some notable exceptions which will be discussed later: see Peltzman (1975), Levitt and Porter
(2001), and Ashenfelter and Greenstone (2004).
5
National Highway Traffic Safety Administration (NHTSA), speeding is a factor in 30
percent of all fatal crashes in the US.5 In 2001, more than 12,000 people died in speedrelated crashes on American roads, at an economic cost to society of more than $40
billion.6 Worldwide, traffic injuries rank second to HIV/AIDS as the leading cause of
ill-health and premature death among the 15-44 age group. Because the number of vehicles per capita is rapidly growing in developing countries, traffic injuries are projected
to be one of the leading public health issues over the next few decades.7
To deter speeding, police in several countries have adopted programs of announced
radar controls that publicize the location and approximate time of operation of radar
controls.8 The data analyzed in this paper were gathered in the Belgian province of
the Eastern Flanders during the years 2000-2003. We have observations on all radar
controls in that time period affecting 6.5 million cars and resulting in 206,146 tickets
issued. The announced controls in the data are observed to rotate in a fairly mechanical
fashion across different sections of the roads and time periods. The police plan the
announcement schedule at least one month in advance, and the timing of announcements
does not appear to be systematically related to speeding propensity. We interpret the
announced controls as crackdowns on particular groups of motorists, those traveling on
the announced section of the road at the announced time. We measure the deterrence
effect of the increased probability of detection by comparing decisions to speed within
the crackdown and non-crackdown groups. Using implications of the theoretical model,
we are able to calculate the effect of increasing the level of resources devoted to speeding
interdiction. In conjunction with value-of-life estimates, our results indicate that at the
current level of interdiction, the marginal benefit, in terms of statistical lives saved, is
close to the marginal cost of interdiction. Importantly, our analysis demonstrates that the
rational theory of optimal interdiction can have not only normative implications (how
the police should behave), but can even generative positive predictions about actual
police behavior (how some police actually behave).
5
6
See Traffic Safety Facts 2001.
Over 80 percent of the economic cost is attributable to lost workplace and household productivity.
See Blincoe et al (2002).
7
See the WHO publication on traffic safety: http://www.who.int/world-health-day/
2004/en/traffic_facts_en.pdf.
8
For example, the Netherlands, Belgium, Germany and Australia.
6
The paper develops as follows. Section 2 presents a theoretical model that we use
to study the conditions under which crackdowns emerge as an optimal policing strategy
when the goal of police is to minimize crime subject to a resource constraint. The
model allows for unobserved heterogeneity in the benefits citizens get from breaking
the law, under the assumption that the police knows the distribution of the unobserved
variable. The model is extended to the case where the police is constrained in terms of
the number of successful interdictions, which is the case in our application to speeding.
Section 3 applies the model developed in Section 2 to data that we obtained from the
Belgian police department. Section 4 provides a discussion and some further extensions
and applications, including tax evasion, drug interdiction, internal compliance, and the
collection of TV licenses in Sweden. Section 5 concludes.
1.1
Related Literature
The idea that deterrence may be improved by focusing interdiction on arbitrary subsets of
the population is present in the literature on racial profiling (see Persico, 2001). Recently,
Lazear (2006) develops a related idea in the context of designing educational tests, where
the question is how much of the test content to reveal to the test-takers ahead of the
test.9 In the nutrition literature, there is a related idea in connection with nutrition
curves. For example, Pratap and Sharma (2002) argue that, in the presence of limited
amounts of food, maximization of family survival and resources may entail an unequal
distribution of nutritional resources, i.e., focusing resources on a subset of the family.
Relative to these strands of literature, the contribution of this paper is (a) to pose a
general policing problem and to characterize the optimal policing strategy; (b) to point
out that crackdowns (in our terminology) allow the researcher to infer the deterrence
effect of policing; and, (c) to empirically illustrate the methodology within a policyrelevant application, speeding.
Our work is also part of the literature on bureaucratic incentives. The most relevant
papers in the economics literature are Prendergast (2001) and Shi (2005). These papers
look at the effect of bureaucratic oversight on policing, with special reference to race
9
Coincidentally, Lazear (2006) also cites speeding as a potential application in developing his argu-
ment, but his and our work are independent. We discuss his paper in detail below in section 4.1.
7
disparities. Our paper also compares the strategies adopted by police under different
incentive schemes, although that is not the main focus.
More broadly, the issue of deterrence in a traffic context is a subset of the vast
literature on crime.10 Of direct relevance to this paper is the literature concerned with
traffic enforcement (speeding, drunk driving, and seat-belt wearing), which is reviewed
by Zaal (1994). Much of the literature on speeding attempts to quantify the effects of
a change in the speed limit on accidents.11 There is also a literature directly concerned
with police enforcement and with estimating the deterrence effect of increased policing
and of greater penalties on speeding.12 Parallel literatures deal with the deterrence effect
of increased policing and greater penalties on drunk driving and on seat-belt wearing.13
Also related is the literature that studies the connection between risk-taking behavior
(speeding, drunk driving) and accidents (see Levitt and Porter (2001)). A paper by
Ashenfelter and Greenstone (2004) uses changes in the speed limit across US states to
estimate the value of a statistical life. They also provide a rich summary of economics
papers in this area.
We acknowledge that our use of the term “crackdown” is somewhat different from
the way the term is occasionally used to refer to increases in interdiction that are not
deliberate randomizations, but rather may be considered exogenous increases in resources
in the sense that they are caused by events unrelated to the crime that is the object
of study. For example, Di Tella and Schargrodsky (2002, 2004), study the effect of
crackdowns on bureaucratic corruption and on crime. Our work complements this line of
inquiry by pointing out that “random” crackdowns can be expected to arise endogenously
as part of the optimal policing strategy.
10
See Becker’s seminal (1968) paper. Also, see Levitt (1997) for a recent study of deterrence in law
enforcement.
11
See for instance Balkin and Ord (2001), Lave (1985, 1989) and the literature cited therein, and
Ledolter and Chan (1996).
12
See e.g. Redelmeier et al (2003), Homel (1988a), Ross and Gonzalez (1988).
13
See e.g. Ross (1984) for drunk driving and Campbell (1988), Peltzmann (1975) for seatbelt use.
8
2
Policing Model
In this section, we characterize optimal monitoring strategies in a model in which the
police minimize crime subject to a budget constraint. One may think of this problem as
originating from an agency relationship in which a principal (a politician, a bureaucrat,
or a high level police administrator) is faced with the problem of giving incentives for an
agent (the police chief) to allocate resources effectively towards achieving some socially
desirable outcome, such as a lower crime rate.14 Citizens, who differ unobservably in
their propensity to commit a crime, choose whether to commit a crime. We assume
that the agent knows the distribution of the propensity to commit crimes across the
population, but the principal only observes the realized crime rate (which depends on
police behavior) and therefore gives the police chief incentives to minimize crime subject
to a resource constraint. We also assume that the police chief receives some amount of
resources (e.g. manpower), and that he can commit to choosing which citizens to police
and with what intensity.
In Section 2.3, we explore several variants of this problem, including one in which the
agent takes into account individual benefits from crime.
2.1
The model
There is population of size 1 that is heterogeneous in the benefit x from committing
a crime, assumed to be unobservable by the police. Let x be distributed across this
population according to a c.d.f. F , and let p denote the probability that the citizen is
monitored. If a citizen commits a crime and is monitored, he is caught and receives
penalty T .15 We assume that p ∈ [0, p], which implies that a citizen can be monitored
with probability no greater than some p ≤ 1.
14
Besides crime reduction, the principal might have other objectives. For example, a principal might
want to allow some citizens who get the highest value to commit crimes. We will explore this point in
Section 4.5.3.
15
Here we assume that the citizen’s utility functions are linear and that committing the crime is a
discrete decision, but in Section 4 we relax these assumptions. Also, for the moment we take T as given.
We will return to this issue in Section 4.
9
A citizen with benefit x commits a crime if
x − pT > 0.
(1)
If a group of citizens is policed with intensity p, the fraction of criminals is
1 − F (pT ) .
The police minimize the crime rate. One possible policing strategy is to monitor every
citizen with the same probability. Alternatively, the police can divide the population into
subgroups and police them at different intensities. Of course, this division only matters
if the citizens know that they are policed with different intensities, so we will assume
that each citizen knows the intensity.16 We denote by µ (p) the size of the group policed
Rp
at intensity p. Because the total size of the population is 1, it must be 0 µ (p) dp = 1.
The contribution of group µ (p) to total crime is µ (p) (1 − F (pT )), and aggregating
over all groups gives the total number of criminals:
Z p
µ (p) (1 − F (pT )) dp.
(2)
0
In this section, we assume that the police’s goal is to minimize the number of criminals.
Alternative specifications of the objective function are studied in Section 2.3, where we
also look at the social planner’s problem.
Let us now turn to the resource constraint. The police chief is assigned some amount
of resources (such as officer-hours). Monitoring a group of size µ with intensity p is
assumed to require police resources in the amount of µ · p. If total resources of P per
capita are available for policing, the aggregate police resource constraint is
Z p
µ (p) pdp ≤ P.
(3)
0
We refer to this constraint as a time constraint. An alternative way of specifying the
constraint on police resources is explored in Section 2.3. The police chooses a probability
measure µ to minimize the number of criminals (2) subject to the resource constraint
(3).
16
In practice, this means that the police must inform citizens of the intensity with which they are
policed.
10
2.2
Analysis
We next provide an intuitive characterization of the properties of the solution to the
police problem previously described. These properties are summarized in Propositions 1
and 2. The propositions are a consequence of Theorem 1, which is stated and proved in
Appendix A.
Let us start by supposing that the solution to the police problem entails policing all
citizens with the same intensity. By the resource constraint, this intensity must equal P .
In terms of our model, this policing strategy corresponds to µ (p) equal 1 if p = P , and
equal zero otherwise. Substituting this choice of µ into the objective function (2) we see
that the number of criminals equals S = 1 − F (P T ) . This situation is depicted in the
left hand panel of Figure 1.
Fraction
speeding
Fraction
speeding
S
S’
1-F (pT)
P
1-F (pT)
p
pL
(Policing
intensity)
P
pH
p
(Policing
intensity)
Figure 1. A crackdown is optimal.
As seen in the right panel of the Figure 1, the number of criminals can be reduced
if resources are allocated differently. If some citizens were policed with intensity pL and
the rest were policed with intensity pH , then it would be possible to bring the number
of criminals down to S 0 < S.
Citizens who are policed with intensity pH are said to be subject to a crackdown.
Figure 1 shows that crackdowns help “iron out” the inward bumps of the function 1 −
F (pT ), thus enabling the police to maintain policing intensity on the efficient frontier.
More precisely, by using crackdowns the police elicit a response function from citizens
that corresponds to the convex hull of the epigraph of (i.e., of the area above) the function
1 − F (pT ).
More formally, this crackdown strategy corresponds to choosing µ (p) > 0 if p =
11
pL , pH , and equal to zero otherwise. The fraction µ (pH ) is optimally chosen to be the
largest possible compatible with satisfying the resource constraint (3), which therefore
reads
(1 − µH ) pL + µH pH = P.
Of course, crackdowns are not always part of the optimal policing strategy. If, for
example, the function 1 − F (pT ) is globally convex, as depicted in Figure 2, then crack-
downs would not be optimal. Even in Figure 1, if P were smaller than pL or larger than
pH , crackdowns would not be optimal. In those cases, the most efficient use of resources
is to police every citizen with the same intensity.
Fraction
speeding
1-F(pT)
P
Figure 2. Crackdowns are not optimal.
When crackdowns are optimal, it is because the function 1 − F (pT ) is not convex.
Given that crackdowns play a “convexifying” role, there is no additional gain in dividing
the population in more than two groups. In fact, given any function 1 − F (pT ), any
point in its convex hull can be achieved as a convex combination of at most two points
in its epigraph. A three group crackdown, therefore, which would entail three different
policing intensities, can achieve nothing more than a two-group crackdown. The following
proposition actually takes this logic a bit further in stating that “generically,” three group
crackdowns are strictly suboptimal.
Proposition 1 Given a homogeneous population with a generic distribution of propensity to commit a crime, the optimal policing strategy involves either monitoring everyone
12
at the same rate, or dividing the population into at most two groups to be monitored with
different intensities.
Proof. This proposition follows from Theorem 1 , which is proved in Appendix A.
17
An extreme form of crackdowns arises when 1 − F is globally concave. In this case,
the convex hull is given by the segment that connects the points (0, 1 − F (0)) and
(p, 1 − F (pT )), which means that for any P we have pL = 0, pH = p. Thus, the optimal
policy entails the use of extreme crackdowns: one group of citizens will be monitored as
intensely as possible, the rest will not be monitored at all.18 This observation gives rise
to the following remark.
Remark 1 If F is convex on its domain, then for any P ∈ (0, p) the optimal policing strategy involves monitoring one group of citizens with maximal intensity, and not
monitoring the others at all.
It is worth pointing out that crackdowns are generally optimal for some P unless f
is monotonically decreasing on its support.
Remark 2 Unless F is concave on (0, pT ) , there exists some P such that the optimal
policing strategy involves random crackdowns.
We now turn to the comparative static result that deals with increases in the police
budget in the presence of crackdowns. The intuition behind the result can be explained
using Figure 1. Suppose that P , the amount of resources available to the police, is
increased slightly. Although the fraction of citizens who are subjected to a crackdown is
now higher because more resources are available, the optimal police strategy still entails
a crackdown with intensities pH and pL . In other words, only the sizes of the two groups
change, but not the intensity with which they are monitored. This simple but important
point, which we note in Proposition 2, is important to being able to use crackdowns to
infer the effect of relaxing the resource constraint.
17
Theorem 1 in Appendix A is a general formulation of the problem we solve here. It allows for
different variations, including those we explore further in the paper. By proving the result for the
general case, all our results follow for each of the specific cases considered.
18
This is the case described in the example in the introduction.
13
Proposition 2 Given total police resources of P , suppose the optimal policing strategy
involves dividing the population into a crackdown group of size µH monitored with intensity pH and a non-crackdown group of size µL monitored with intensity pL . Consider an
increase in total police resources to P̃ ∈ (P, pH ). In the new optimal strategy the crack-
down group is larger than before, (i.e., µ̃H > µH and thus µ̃L < µL , the non-crackdown
group is smaller), but the intensities with which the two groups are monitored remain
unchanged (they are still pH and pL ).
Proof. This proposition follows from Theorem 1 , which is proved in Appendix A.
Proposition 2 provides a way of forecasting the deterrence effect of an increase in
police resources. Crucially, the approach does not require knowledge of the shape of the
function 1 − F (pT ). Refer again to Figure 1. Graphically, increasing P results in the
crime rate S 0 sliding down along the shaded segment. The slope of the shaded segment,
therefore, determines the degree to which crime decreases as resources increase. This
slope can be calculated based on the formula
[1 − F (pH T )] − [1 − F (pL T )]
.
pH − pL
Multiplying this slope by Pe − P provides a way of estimating the expected decrease in
crime due to a hypothetical increase in police resources from P to Pe. Thus,
∆Crime (crime rate|pH ) − (crime rate|pL )
.
=
∆P
pH − pL
The terms in the numerator on the right-hand side (the crime rates with and without
crackdown) as well as those in the denominator (the intensity of monitoring) would be
observable in most applied settings in which crackdowns are observed.
To see why observing crackdowns is necessary to carry out this computation, consider
the³ no-crackdown
primitives depicted in Figure 2. We are interested in forecasting 1 −
´
F PeT , the crime rate after the increase in the budget. Absent any information on
³ ´
the shape of the function 1 − F (pT ), there is no way to compute 1 − F PeT based on
the available information, which is only the knowledge of 1 − F (P T ), the initial crime
rate. This is why most of the literature on deterrence focuses on identifying sources of
exogenous variation in P , which allows one to trace out (or at least locally approximate)
14
the function 1 − F (P T ) as P varies. However, in the presence of crackdowns there is
no need to identify sources of exogenous variation in P to identify the deterrence effect.
One can think of crackdowns as being a case where exogenous variation in P arises as
part of the optimal policing strategy.
If, in addition to observing crackdowns, one also has access to exogenous variation in
total police resources P , then Proposition 2 yields a testable implication of the model.
The implication is that, as P increases (between pL and pH ), the optimal monitoring
intensity should not change, but the size of the group subjected to crackdowns should
increase. In Section 3.4, we consider this implication in the context of speed interdiction.
2.3
Constraint on successful interdictions
In the empirical application of section three, the Belgian police are given a constraint
on the total amount of tickets that they are allowed to write, rather than a constraint
on manpower. In this Section we adapt the model to the problem faced by the Belgian
police.
As before, we assume that the goal of the police is to minimize the crime rate subject
to a constraint. The resource constraint is now given in terms of the number of successful
interdictions rather than in terms of time resources. That is, monitoring criminals has a
cost to the police, but monitoring honest citizens is costless. This captures environments
in which interdiction is cheap relative to the cost of processing violations. This happens to
be the case in our speeding application of Section 3, where the police are administratively
restricted in the number of tickets that they can issue in a year.19
The term (1 − F (pT )) · p represents the number of successful interdictions from a
population that is policed with intensity p. To capture the constraint on successful
interdictions we modify the constraint (3) to read
Z p
µ (p) (1 − F (pT )) p dp ≤ C.
(4)
0
The police minimizes expression (2) subject to the constraint (4).
19
This makes sense because detecting speeders with automatic radar machines is almost costless
relative to processing a traffic ticket. More on this in Section 3.
15
To rule out trivial cases where the resource constraint is not binding, we assume that
C is such that the police could not afford to monitor everyone with maximal probability.
This assumption, which will be maintained throughout, is
Assumption C < (1 − F (pT )) p.
The present problem shares a key formal similarity with the benchmark model: both
programming problems are linear in µ. As a consequence, even though constraint (4) and
constraint (3) are quite different, Propositions 1 and 2 continue to apply. In constraint
(4), for example, the kernel of the integral is not necessarily increasing in p: higher
monitoring intensity does not necessarily entail more successful interdictions. Yet, at
the optimal solution, one can show that more resources (successful interdictions) must
be expended per capita on the crackdown group than on the other group. One can also
show that whenever crackdowns are optimal in the benchmark model for all values of P ,
then they are also optimal when police are ticket constrained. These results are collected
in the following proposition.
Proposition 3 Consider a monitoring problem in which police minimize crime under
the constraint that successful interdictions not exceed C. Then:
a) The optimal monitoring strategy involves either monitoring everyone at the same
rate or dividing the population into at most two groups, which are monitored at different
intensities.
b) Given C, suppose the optimal policing strategy involves dividing the population
into a crackdown group of size µH monitored with intensity pH and a non-crackdown
e ∈
group of size µL monitored with intensity pL . Consider an increase in C to C
(C, (1 − F (pH T )) · pH ). In the new optimal strategy the crackdown group is larger than
before, (i.e., µ̃H > µH and thus µ̃L < µL , the non-crackdown group is smaller), but the
intensities with which the two groups are monitored remain unchanged (they are still pH
and pL ).
c) At the optimal monitoring strategy, the expected number of successful interdictions
per capita (the probability that there will be monitoring multiplied by the fraction of
motorists who speed) is larger in the crackdown group than in the non-crackdown group.
d) Crackdowns are optimal for all values of C if they are optimal in the benchmark
model for all values of P . The converse is not true.
16
Proof. a), b): see Theorem 1.
c) Suppose not. Then if one perturbed the optimal strategy by shifting a small mass of
citizens from the non-crackdown group to the crackdown group, the resource constraint
would continue to be satisfied and the crime rate would decrease. This contradicts
optimality of the original strategy.
d) Let P denote the maximal feasible policing intensity when all motorists are policed with the same probability. This is the monitoring intensity that minimizes crime
among all feasible non-crackdown strategies. For future reference, observe that feasibility
implies C = P (1 − F (P T )). Consider now the ancillary problem which is to minimize
crime subject to the constraint (2). Let µL , pL , µH , pH denote the optimal crackdown
probabilities in the ancillary problem. By definition, this crackdown policy generates a
lower crime rate than if all citizens were policed with intensity P . We now show that the
same crackdown strategy is feasible in the original problem. This will prove that equal
policing is dominated by crackdowns in the original problem.
To verify feasibility in the original problem, write the following chain of inequalities:
C = P (1 − F (P T ))
= [1 − F ((µL pL + µH pH ) T )] (µL pL + µH pH )
≥ µL pL (1 − F (pL T )) + µH pH (1 − F (pH T )) ,
where the inequality reflects the concavity of the function x [1 − F (xT )]. This function
is concave because F is convex, which we know because crackdowns are optimal for all
values of P in the ancillary problem (refer to Remark 1).
Part d) of the above proposition suggests that crackdowns can be optimal when the
police are ticket constrained even in cases where they are not optimal in the benchmark
model. The intuition for crackdowns when the police faces constraint (4) is as follows.
In a crackdown, the high interdiction group commits little crime, while the group that
is more prone to committing crime is rarely policed. This tends to reduce the number of
tickets that are written relative to the case in which both groups are policed at the same
rate. Thus, besides helping satisfy the objective function, engaging in crackdowns has
beneficial effects on constraint (4). The second effect was not present in the benchmark
problem.
17
Part (b) of Proposition 3 yields a useful formula for computing the effect of an increase
in police resources on the crime rate. Because this formula will be used in the empirical
e equals
work in section three, we derive it here. The crime rate given C
µ
e (pL ) · (1 − F (pL T )) + µ
e (pH ) · (1 − F (pH T ))
(note that there is no tilde over the p’s in light of Proposition 3 part (b)). To obtain the
change in crime, subtract from this expression the analogous expression when resources
equal C. This yields
(5)
∆Crime
µ (pH ) − µ (pH )] (1 − F (pH T ))
= [e
µ (pL ) − µ (pL )] (1 − F (pL T )) + [e
= [e
µ (pH ) − µ (pH )] (F (pL T ) − F (pH T )) .
The optimal policing strategy µ
e must meet the budget constraint, and so
e = (1 − µ
eH (1 − F (pH T )) pH .
C
eH ) (1 − F (pL T )) pL + µ
Isolating µ
eH yields
µ
eH =
e − (1 − F (pL T )) pL
C
.
(1 − F (pH T )) pH − (1 − F (pL T )) pL
The optimal policing strategy before the increase in resources must also meet the budget
constraint, and so
µH =
C − (1 − F (pL T )) pL
.
(1 − F (pH T )) pH − (1 − F (pL T )) pL
Substituting into (5) we get
´·
³
e−C
∆Crime = C
¸
F (pL T ) − F (pH T )
(1 − F (pH T )) pH − (1 − F (pL T )) pL
¸
·
(crime rate|pH ) − (crime rate|pL )
.
= ∆C ·
(crime rate|pH ) · pH − (crime rate|pL ) · pL
(6)
All the terms in the right-hand side brackets are observable when the resource level equals
C. Thus, the decrease in crime rate due to an increase in resources can be computed even
without observing any variation in the data in the level of police resources. In Section 3
this slope is calculated in the context of highway speeding interdiction.
18
3
Empirical Application: Speeding Interdiction
In this section we apply our theoretical model of policing to study speeding interdiction
in Belgium. As discussed in section one, speeding interdiction is an important policy
question in its own right, because traffic accidents are a leading cause of death and
disability worldwide.20
Our goal in this section is to estimate the deterrence effect of
resources devoted to speeding deterrence by directly applying the theoretical framework
developed in section two.21 Then, in conjunction with value-of-life estimates taken from
the literature, we consider whether the marginal benefit, in terms of statistical lives saved,
warrants the marginal cost. Our estimates indicate that the current level of interdiction
is close to socially optimal.
3.1
The environment and the data
Our data come from the administrative records of the Belgian police department. In three
Belgian provinces (Eastern Flanders, Liege and Luxembourg), the police puts extensive
effort into publicizing announced radar controls, through different media that include
newspapers, radio, internet, local stores and restaurants.22 Our analysis samples cover
the province of Eastern Flanders, which has two major highways and one minor highway,
each of them connecting to the city of Gent.23 The two major highways are divided into
four sections: A14-North, A14-South, A10-East, and A10-West. In total there are 5
20
In 1990, for example, traffic accidents represented the fourth leading cause of loss of DALYs (dis-
ability adjusted life years) in developed countries. Worldwide, accidents were the third cause of loss of
DALYs for ages 15-44. By comparison, war was only the seventh leading cause for those ages.
21
The application of the theoretical model to speeding interdiction has the advantage that the issue
of incapacitation does not play a significant role.
Speeders do not often receive prison sentences,
although they may temporarily get their license revoked. Crime rates can be reduced by deterring
potential criminals or by incapacitating them, and the models we examined in section two dealt only
with deterrence.
22
The controls are announced, for example, on the website: http://www.federalepolitie.be.
23
The East-West highway A10 connects Gent with Brussels to the East and with Bruges to the West
and the North-South highway A14 connects Gent with Antwerp and the Dutch border to the North,
and with the French border to the South. Both highways are of approximately equal length and cover
around 60-65km within the province of Eastern Flanders. The third highway, R4, is a short section of
highway connecting Gent with its port to the North.
19
sections, each of which is of roughly equal length (30-40km). The province has two radar
control machines that can be placed along roads or highways to record the speed of
drivers passing along that road and to take photographs of cars that are speeding, which
are then issued tickets.24
An announcement merely says that a section of one highway (for example, A14North) will be subjected to increased monitoring in a time period (for example, between
the hours of 6am-12). The announcement does not specify the direction of the road
on which the machine is placed, nor of course its exact location. In fact, the police
generally hide the position the radar machine, so as to avoid the possibility that drivers
may slow down in the proximity of the machine and then pick up their speed again.25
The announced controls in the data are observed to rotate in a fairly mechanical fashion
across different sections of the roads and time (more on this later). On any given day,
the police either make no announcement or, when they announce, they keep one of the
machines in reserve to possibly monitor some other section in an unannounced way.
The data we use for the empirical analysis record the date, time of day, and location of
the machine and whether the radar control was announced as well as information directly
recorded by the machine, such as the number of vehicles passing by the machine, the
fraction of cars and trucks that were driving in excess of the speed limit (the limit differs
for cars and trucks) and the fraction of vehicles exceeding the speed limit by 15 km/h.26
24
Radar control machines record speeds and take photographs of speeding vehicles. The license in-
formation obtained from the photo and information recorded on the speed is used to issue the tickets.
If the driver passes a radar machine while exceeding the speed limit by a certain threshold the probability is close to one of receiving a ticket. It is not equal to one, because, rarely, sun glare makes the
photo unreadable. The radar control machines are mobile and are typically moved to several locations
throughout the day.
25
Only in rare cases (less than 1% of our data) is the radar machine not hidden. Even then, it does not
seem to be easily detectable by drivers: we verified that the probability of speeding does not decrease
in those cases. The same finding was independently reported to us by the police, based on experiments
they have done positioning a second radar machine several kilometers after the first radar.
26
A major speeding violation is defined in the law as traveling at a speed of 10km/h or more over
the speed limit. In addition, the margin of error of the radar is ±3%, or 4km/h at the maximum speed
on highways, which is 120km/h. It is police discretion when to issue a ticket for major violations, and
currently, police considers a major violation traveling at a speed of 136km/hr or higher on highways.
During the time period covered by our data, tickets were not issued systematically for minor violations
20
In what follows, we will use the following terminology. A monitoring event refers
to each of the possible time-space combinations in which the police sets out the radar.
There are a total of 5475 monitoring events: 3 time periods (6am-12, 12-6pm, and 6pmmidnight) per day for 365 days, and 5 comparable, equidistant sections of road (2 on
A10, 2 on A14, and 1 on R4). Our choice of the time period is natural given the police
announces controls within those 6 hour time periods, and for comparability, we use the
same unit of account for unannounced controls. Of course, given random controls, the
actual duration of controls in any 6 hour period is shorter. We refer to a monitoring
interval as the actual duration of monitoring during any 6 hour monitoring event. In
the data there is variation in the monitoring intervals, with an average of 3 hours.
During announcement events, drivers know that monitoring can happen on the specified section during part of the 6 hour time period and in either of the two directions.
Within our model, this allows the driver to form beliefs about the likelihood of being
monitored. When no announcement is made, the driver knows that there can nonetheless
be monitoring and, within our model, forms a different belief about the probability with
which this happens.
Police objective and constraints. The police department explicitly states that its
goal in issuing tickets is to deter speeding. Maximizing the revenue from traffic tickets is
not an objective, and in fact, the police do not get to keep the revenue from the tickets
they write. Through conversations with the police, we learned that they face a binding
constraint on the total number of tickets. The primary cost of issuing a speeding ticket
is the administrative processing cost, which is estimated to be about US $0.50.
The
police are given a total budget at the beginning of the year (allocated by the legislature),
so they know how many tickets can be issued during the year within this budget.27 The
(over the speed limit, but less than 136km/h) as a result of the radar data collected.
27
This institutional arrangement raises at least two questions. 1. Why not let the police choose the
number of tickets and allow them to keep a fraction of the proceeds? The likely reason is that politicians
and the police want to avoid the appearance of a monetary incentive to write more tickets because the
public does not want police who are revenue maximizers. In fact, the statutory objective of the traffic
police is to minimize speeding, and not to maximize fine revenues. 2. Why not relax the constraint
on the number of tickets? Our calculation (Section 3.5 below) does not indicate a clear welfare reason
for wanting to relax the constraints, so it is not immediately clear that the politicians are imposing an
inefficiency by limiting the number of tickets.
21
budget determines in large part the total number of ticketed speeders per year, that is
reported in Table 1a. To avoid issuing too many tickets, police do not make use of the
radar control machines every day. On days when no announcement is made, police may
or may not use the machines. On days with announcements, police make use of at least
one machine on the announced road and may also use the other machine on the same or
another road unannounced.
Following organizational reforms within the police, in 2002 and 2003 there was a sharp
increase in the number of tickets that the police were allowed to issue on highways.28 The
size of the budget constraint more than doubled, from 33,951 tickets in 2000 to 78,136
tickets in 2002. Based on conversation with the police, this reallocation of funds was
not triggered by any perceived change in the motorists’ propensity to speed, but rather
resulted from broader organizational changes.29 We will therefore treat this change in
the police budget as exogenous, and we will use this change as an additional way of
examining support for the model.
Monitoring Policy. The monitoring policy is determined at least one month in advance
of the actual radar control. For this reason, the officer who schedules the times and
locations of announced and unannounced controls does not react to short term changes
in circumstances, such as unpredictable weather conditions. The planned radar controls
appear to be always implemented. Table 1a shows the number and percentage of vehicles
subject to announced and unannounced radar control on the three major highways for
years 2000-2003. Table 1b shows the number of drivers issued speeding tickets. Table
1c reports the number of monitoring events on each road, where a monitoring event is
defined as the machine being placed on a road and recording information for some time
interval. Highway A14 has the highest level of monitoring, followed by A10 and then
the shorter highway, R4. Table B1 in the Appendix tabulates the number of announced
and unannounced monitoring events by month of year. There is no systematic pattern,
28
Prior to the reforms, the state police monitored both highway and nonhighway roads. After the
reforms, there was a change in the jurisdiction so that state police monitored only highway roads and
local police monitored nonhighway roads, and the number of tickets that the state police could issue on
highways increased. We do not know what happened to the level of resources on non-highway roads,
as they are not recorded in our dataset, which comes from the state police department.
29
This change occurred, in part, because resources previously used to monitor both highways and
some smaller roads were from 2002 on earmarked for highways only.
22
except that monitoring is more frequent in the month of December. Table B2 tabulates
monitoring and announcement events by day of the week and shows that monitoring is
more frequent on weekend days in 2000 and 2001.
Driver’s avoidance. One potential concern in applying our model to the data is
whether drivers who hear the radar control announcements can select an alternate route
to avoid detection, which would mean that the speeding response of people who choose
to remain on the announced road could no longer be compared to the response of drivers
in the absence of the announcements. Because our data pertain only to major highways,
the potential problem of route selectivity is mitigated. If a motorist wants to avoid a
highway with announced radar controls, she will necessarily have to take country roads,
with relatively low speed limits (between 50 and 90 km/h) and with traffic lights. On
the basis of time cost, a driver should prefer to take the highway rather than a country
road, even with the announced controls. Aside from spatial avoidance, one might be
concerned about temporal avoidance: anticipating or postponing one’s travel to avoid
monitoring. However, the length of an announced monitoring interval is 6 hours. A
driver whose ideal driving time is in the middle of this interval would have to anticipate
or postpone his travel by 3 hours. Although there will be some drivers who can engage
in some avoidance with little change to their ideal driving schedule, that only applies to
those who are ideally driving at the very beginning or very end of the 6 hour interval. In
fact, we computed that if monitoring induces a driver to reduce his speed by 20 km/h,
then the time loss on the 40 km length of our section is about 3 minutes. According to
this calculation the concern about temporal avoidance is limited to a small fraction of
the monitored population.
3.2
Constructing the variables and fit to the model
The theoretical model presented in Section 2 assumed that individuals are heterogeneous
in their propensities to speed and that the heterogeneity is unobserved by the police.
In reality, the population driving on the road may vary in some observable as well as
unobservable ways. For example, police may be aware that speeding is more common
during certain times of the year or on weekends We denote the set of characteristics that
are observable to police by Z and assume that the theoretical model applies conditional
23
on a set of observables Z. Table 2 reports the summary statistics of the variables used
in the analysis.
Estimation of the expected probability of monitoring. Recall that in the theory
pL and pH represent drivers’ perceived probabilities of being caught speeding on unannounced (low) and on announced (high) monitoring days. On announcement days, police
monitor the announced road with probability one. A typical announcement specifies the
section of road being monitored and the length of time (for example, one hour in the
morning). The driver’s perceived probability of being caught speeding is typically less
than one, because police do not monitor the entire length of time of the announced control and they only monitor one of the two driving directions. We assume that drivers
form expectations about the probability of being subject to monitoring, and that they
may use characteristics of the day in forming their expectations. We therefore allow
pL (Z) and pH (Z) to depend on Z. For example, a driver may know that the police tend
to do more monitoring on weekends or on holidays.30
On days with announcements, monitoring is announced within three separate time intervals, corresponding roughly to morning (6am-12pm), midday (12pm-6pm) and evening
(6pm-12am). We therefore divide each day of the year into three potential monitoring
intervals. For each year, we therefore have a total of 5475 monitoring events: 365 days,
3 time intervals per day, 5 sections of road (2 on A10 and A14 and 1 on R4). Let m be
an indicator for whether any monitoring takes place during an interval and let a indicate
whether the monitoring was announced. On announcement days, monitoring definitely
occurs, so Pr(m = 1|a = 1, Z) = 1. On days without announcements, drivers form a belief about the expected probability of being monitored, potentially taking into account
factors such as the day of the week, the month of the year, whether the particular road
was recently subject to monitoring, whether there has been an announcement on another
road or whether it is a holiday. We estimate the predicted probability that a road is subject to monitoring (Pr(m = 1|a = 0, Z)) by a logistic regression, using as the estimation
30
Because the police hide the location of the radar machine, a driver’s belief about the probability
of being monitored should be constant along a section of highway. In reality, not all motorists travel
along the entire section–some make shorter trips. We will show below that, for our purposes, there is
no loss in generality in treating shorter trips in the same way as longer trips.
24
sample all potential monitoring events in which there was no announcement.31
On both announced and unannounced monitoring days, to arrive at a perceived
probability of being monitored, drivers have to form an expectation about the number of
hours that the monitoring will take place. Using data on the actual length of monitoring
during the event, we estimate a regression in which the dependent variable is numbers of
hours spent monitoring. The independent variables indicate the day of week, month of
year, holiday, and whether there were recent monitoring activities on that road. Using
these ingredients, the predicted probability on announcement days, pH (Z), is obtained
by
H
|m = 1, a = 1, Z),
TH
where H represents the number of hours of monitoring and TH is the total time of the
pH (Z) = 0.5 · E(
announced monitoring event. The multiplication by 0.5 accounts for drivers’ uncertainty
about which direction of the road will be subject to the monitoring.
The corresponding probability on unannounced days, pL (Z), is
pL (Z) = 0.5 · Pr(m = 1|a = 0, Z) × E(
H
|m = 1, a = 0, Z).
TH
pL (Z) is typically much lower than pH (Z), because Pr(m = 1|a = 0, Z) is much less than
1.32
Appendix B, Tables B3 and B4, report the estimated regression coefficients for the
estimation of the predicted probability of monitoring on unannounced days and for the
predicted length of time spent monitoring. As can be seen in Table B3, there are few
systematic predictors of monitoring. It appears that the roads are more likely to be
monitored unannounced if there was some unannounced monitoring in the previous week.
31
This estimation includes all potential monitoring events (365 days of the year times 3 intervals per
day). The data only record events in which monitoring took place, but we could infer the characteristics
of the events during which there was no monitoring (day of week, time of day, month of year, holiday,
whether there was recent monitoring).
32
For days without announcements, we set TH = 16, because we do not observe in the data any
monitoring during nighttime. For roads A14 and A10, we make an additional adjustment (multiply by
0.5) to the estimated probabilities pL (Z) to take into account drivers’ uncertainty about which section
of road is subject to the monitoring. On announcement days, drivers learn the section of road from the
announcements but on unannounced days they do not know the section of road where monitoring will
take place. On the shorter road R4, there is only one section so no further adjustment is necessary.
25
Table B4 shows that few of the conditioning variables have any predictive power in
explaining the length of time police spend monitoring. There is some indication that
the time spent monitoring differs across quarters of the year, but there is no systematic
pattern across all highways.
Finally, a note about the interpretation of p. In the theory, p represents the probability
that a motorist traveling along an entire sector is monitored, equation (1) must be
interpreted as a “per sector” equation. Thus, we take x to represent the “per sector”
benefit of speeding.33 A motorist who traveled only a fraction m of a sector would speed
if mx − mpT > 0, or equivalently, if x − pT > 0. Thus, the motorist’s decision problem is
invariant to the fraction of the sector travelled. This formulation allows us to aggregate
trips of different lengths, which is convenient because we do not observe the length of
each individual trip.
Computing µ. µ(pH ) represents the fraction of monitoring effort devoted to high intensity interdiction (announced monitoring in our model). It is calculated as follows: in
the denominator are all the 5475 possible monitoring events (365 days, 3 time blocks
per day, 5 sections of road); in the numerator are the number of announced monitoring
events (then µ(pL ) = 1 − µ(pH )). The value of µ(pH ) are reported in Table 1c. For
example, in 2000, µ(pH ) = 66/5475 = 1.21%.
Crime rate. Our theory requires us to compute F (pL T ) and F (pH T ), the fractions of
speeders with and without a crackdown (see equation (6)), which we can obtain directly
from the data.
3.3
The randomness of crackdown
One of the premises of our theory is that crackdowns are random. That is, the theory assumes that the difference in monitoring intensity between crackdowns and noncrackdowns (pH and pL ) reflects a non-convexity in F, rather than the police adapting
their monitoring intensity to several convex F ’s, reflecting shifts in the propensity to
speed that are observable to the police but unobservable to us. Figure 3 shows that the
distributions of the pH ’s and pL ’s are bunched in proximity of their respective means
33
We can think of x as reflecting a time benefit from speeding over some interval, and we would expect
value of time to differ across individuals.
26
and the two distributions have literally no overlap (there are no monitoring events with
intensity between 0.05 and 0.15). The distribution of unobservables would have to be
very peculiar to generate this “bimodal” pattern of monitoring intensities. That is, we
would require the existence of a characteristic of drivers on a particular road segment
that observed by the police but unobserved in our data, which sharply and discontinuously shifts their propensity to speed in a way that explains the bimodality. To the best
of our knowledge from conversations with the police, there is no such characteristic and
the bimodality is better explained by a non-connvexity in F.
There is, moreover, evidence that the police do not attempt to strategically condition
announcements on factors that might influence the propensity to speed. First, the police
commit to their announcement schedule at least 1 month in advance, and in so doing they
forgo the opportunity to fine-tune their announcements to factors that might influence
speeding (such as unseasonable weather, road conditions or traffic density). Second, the
announcement schedule is a fairly mechanical rotation, and in personal communication,
the police indicated that they were attempting to make the announcement schedule as
random as possible.
There is thus ample anecdotal evidence that in our context crackdowns are random.
In applications where such anecdotal evidence is absent (see Section 4.5 below for a
description of alternative applications of random crackdowns) our theory can nonetheless
offer a plausible foundation for empirical work provided one can establish support for
the randomness of crackdown.
To further explore this question, Table B6 reports the estimated coefficients from a
logistic regression of the probability of an announcement. The independent variables are
characteristics of the day, such as whether it is a holiday and the quarter of the year,
and information on recent monitoring events. We also include as a potential predictor
variable the percentage speeding on the same road on an unannounced day, as a proxy
for the underlying speeding propensities of motorists on that road.34 Almost all the
variables, including the speeding measure, do not predict the announcements, which is
generally supportive of the announcements being random.35
34
An anonymous referee pointed out this strategy for testing the randomness of crackdown. We are
grateful for this suggestion.
35
We expect the year effects to be statistically significant, given that the operating budgets are annual
27
Even if there seems to be no clear evidence of targeted announcements our estimation
strategy allows for potential systematic variation in monitoring behavior based on a set
of observables Z that police and drivers might plausibly use.
3.4
Examining support for the model
Ideally, we would like to be able to estimate the empirical counterpart of the distribution
of propensities to speed, as depicted in Figure 1, and look for a concave part in order to
justify a crackdown. However, if the theory is correct, then we should observe only two
points on the function F (pL T ) and F (pH T ), and there will not be any variation in the
data to trace out the shape of the function. In the absence of a direct test on the shape
of the distribution of propensities to speed, we turn to our model, and in particular to
Proposition 3 for other kinds of predictions that can be used to examine the empirical
support for the model.36
Implication #1: Partitioning into Two Groups One implication of the theory
(Proposition 3 part (a)) is that the optimal policing scheme partitions the population
in at most two groups. The optimal monitoring strategy involves either monitoring
everyone at the same rate or dividing the population into at most two groups that are
monitored at different intensities. The fact that police announce crackdowns on some
roads on some days naturally gives rise to three different groups: (i) highway sectors with
crackdowns; (ii) sectors without crackdowns on days in which crackdowns are announced
on other sectors, and (iii) sectors on non-announcement days.
[ Figure 3 ]
and the number of announcements increases over time.
36
In order to estimate the empirical counterpart of the distribution of propensities to speed, one
might look for variation in the day-to-day monitoring probabilities (the x-axis in Figure 1) that is (a)
exogenous to the propensity to speed, and (b) anticipated by drivers. Unfortunately, this is difficult.
Weather and traffic density, for example, are sources of variation, but they are likely to affect the
drivers’ underlying propensity to speed, thus shifting the curve in Figure 1. Thus, these are not sources
of exogenous variation. In fact, we account for these factors in the empirical analysis by introducing
them as conditioning variables. Other, more subtle sources of variation in the observed monitoring
probabilities are unlikely to be anticipated by drivers.
28
Figure 3 plots the histogram of the probabilities with which police monitors drivers
(the plot refers to road A14; the histograms for A10 and R4 are similar and available upon
request). The columns of the figures refer to year 2000, 2001 and 2002, and the mean
of the probabilities in each years is shown in the x-axis label. For each year, the figure
in the first row displays the distribution of the probabilities if there was an announced
control (i.e. of pH (Z)). The mean of this distribution is around 25% because, during
an announced control, the police actually has the machine out about 3 hours within an
announced 6 hour time window, on 1 of 2 possible driving directions, so
3
6
×
1
2
= 0.25.
The middle row depicts the distribution of probabilities if there was no announcement
on that road (A10), but there was an announcement simultaneously on another road.
The bottom figure corresponds to the case without announcements on that road (A10)
or on any other road. Groups (ii) and (iii) are monitored with similar intensities, and
with sharply lower intensity than group (i), which is consistent with the prediction of the
theoretical model. However, a statistical test of equality of the estimated probabilities
in groups (ii) and (iii) rejects the hypothesis at 5% significance level. The probability
of being monitored on a given road is slightly less when there is announced monitoring
on another road.
Implication #2: Effect of an Exogenous Increase in Resources on Monitoring
Probabilities The theory (Proposition 3 part (b)) predicts that with an increase in the
number of tickets there will be an increase in announced controls (µH increases, with a
corresponding decrease in µL ) but no change in the monitoring probabilities. In 2002
and 2003, there were large increases in police resources that were arguably exogenous
with respect to the speeding propensities of the drivers. The predictions of the model
that the probabilities remain stable and that announcements occur more frequently seem
to be borne out in the figures. Table 1c shows a very significant increase in µH in 2002,
to 6.15%, more than five-fold compared to 2000. A simple comparison of the histograms
in columns 1, 2, and 3 in Figure 3 reveals that as the number of tickets issued nearly
doubled in 2002, the number of vehicles subject to monitoring increased dramatically but
the monitoring probabilities remained roughly the same. The stability in the probability
of monitoring across years can also be seen in Table B3, where the estimated year effects
are generally not significant determinants of the probability of monitoring, as well as in
Table B5, which shows the average predicted probability of monitoring for each of the
29
years.
Implication #3: More tickets during crackdowns The model predicts that the
expected number of successful interdictions per capita is larger in the crackdown group
than in the non-crackdown group (Proposition 3 (c)). The expected number of successful
interdictions per capita equals the probability that there is monitoring multiplied by
the fraction of motorists who speed.
On days with announcements, the probability
of monitoring increases substantially. If the fraction speeding went down enough to
offset the increased probability of monitoring, then it is conceivable that the number of
successful interdictions could go down. The data are, however, consistent with the model
and the number of successful interdictions per capita increases with announcements. For
example, on highway A10 the expected number of tickets per capita on announcement
days is 0.505% whereas on unannounced days it is 0.026%.37 Likewise on A14 (1.010%
and 0.083% respectively) and R4 (1.041% and 0.038%).
This test of the model is
arguably weak, because the only way the test could fail is if the announcements had a
great deterrence effect and motorists were about ten times less likely to speed during
announced periods.
Overall, we conclude that the data broadly support the key predictions of Proposition
3. We next use the model to infer the deterrence effect of resources devoted to speeding
interdiction.
3.5
The deterrence effect of announced controls
The main goal of our empirical analysis is to estimate the deterrence effect of a change
in the number of tickets issued by the police. The simplest approach would be to measure deterrence by the change in total crime before and after a reform that exogenously
changes the level of resources. We do not, however, have total crime in our data, which
only records information during monitoring events, so this simple approach is not feasible.38 Our model provides an alternative way of estimating the deterrence effect that
37
This calculation is based on estimates of pL and pH and of the percentage speeding with and without
announcements that are estimated by the method described in the next section and are reported in Table
4 (the first column and the column labeled (3)).
38
Following this simple approach would require us to know total crime, i.e. the number of vehicles on
the road and the speeding rates on all days and times of day when there was no monitoring, which is
30
only requires knowing the speeding rates during crackdown and non-crackdown periods.
The theoretical model implied that relaxing the budget constraint should lead to
an increase in the number of announced controls. Drivers who take to the roads on
announcement days are subject to a higher probability of being caught speeding and are
therefore viewed as the group subject to a crackdown. Here, the criteria by which the
crackdown group is distinguished are time and day of traveling on the road.39 To estimate
the deterrence effect, we compare the speeding response on days with announcements
(days with pH ) to the speeding response on no announcement days (with pL ). The
maintained assumption is that announcements (crackdowns) are random conditional on
the observed covariates. The theoretical model of section two assumed that monitoring
intensity is the only observable factor affecting speeding decisions, but in reality other
likely important factors are weather conditions, traffic density, day of the week, month of
the year, time of the day and whether it is a holiday. We therefore include those additional
variables as potential determinants of speeding decisions. To examine sensitivity to the
set of included covariates (Z), we report results with and without conditioning on the
additional covariates.
We estimate a discrete choice logistic model for drivers’ decisions to speed, where the
speeding decision is assumed to depend on the probability of monitoring and on the other
covariates. Table 3 presents the estimated coefficients for three different specifications. In
specification (1) the speeding decision is assumed to be a function solely of the monitoring
probability. Specification (2) adds the conditioning variables that may also be relevant to
the speeding decision: indicators for different levels of traffic density, an indicator for poor
something we do not know. In the absence of this information, we could impute the number of vehicle
and the speeding rates on the days with no monitoring. A difficulty with this imputation is that there
are some observables that we do not know for days when there was no monitoring, such as traffic density
and weather. With our model, we obtain the same objective, i.e. a way to get from the two crime rates
to the deterrence effect. In addition to the fact that we do not need to know the total level of speeding
vehicles, the model also allows us to examine policy changes that are not observed, i.e. any change in
tickets different from the increase of 32,872 observed in 2002-2003.
39
The observed policy of rotating announcements across space and time is not the unique optimal
policy in our theoretical model. Alternatively the police could focus all resources on one particular road
(for example always monitor one section of A10 intensely and always monitor all other sections with
low intensity). We believe that a plausible reason (outside our model) for rotating announcements is to
allocate the burden of interdiction “fairly”.
31
visibility on the road, an indicator for morning and evening rush hour weekday traffic, an
indicator for whether the day is a holiday, and fixed effects for days of week, months of
year, and year. Specification (3) includes the same set of conditioning variables, but the
speeding decision is now assumed to depend only on whether there is an announcement,
without taking into account the information contained in the length of the monitoring
event. This would be the appropriate specification if driver’s expectations about the
probability of being monitored only depend on whether an announcement was made and
not on other day-specific factors.40
As seen in Table 3, speeding decreases during announcement periods and is a decreasing function of the probability of monitoring. This result is robust to the inclusion
of conditioning variables, although a comparison of the estimated coefficients across
specifications shows that the estimated deterrence effect is smaller in the specifications
that include the covariates. Controlling for covariates especially affects the estimated
coefficient associated with the probability of monitoring on highway R4. As expected,
individuals are more likely to speed when traffic density is lower. Speeding also tends to
be higher during weekday rush hour times, on holidays, and on Sundays.
Using the estimated coefficients from Table 3, we estimate for each person the probability of speeding, evaluated at pH and pL values that are set at the average over all
the observed values. The average pH and pL values for each road are reported in the
first column of Table 4. Column (1) of Table 4 reports the average predicted decrease in
speeding on each road that can be attributed to the announcements for the same three
model specifications that were shown in Table 3.41 As noted above, the estimated deterrence effects are smaller when additional covariates are included in the specification. We
focus on the coefficients that include the covariates (reported in columns (2) and (3)),
because they are likely to be important determinants of speeding decisions. On highway
A10, the estimated coefficients imply that announcements reduce the fraction of drivers
speeding on average by 5.0-15.5%. For highway A14, estimates range from 8.0%-18.7%,
40
The results in the third column are robust to any potential misspecification in the model for the
probability of monitoring.
41
That is, we obtain a predicted decrease in speeding for each driver and then take the average over all
drivers. The predicted decrease in driver-specific in the specifications where pL (Z) and pH (Z) depend
on covariates Z, which are driver-specific (such as day and month of travel).
32
and, for highway R4, from 1.9%-8.6%.
These estimates can now be used to compute the deterrence effect of a hypothetical
increase in police resources in the form of an increase in the number of tickets issued.
The effect of an increase of 10, 000 tickets is reported in Table 4 for each of the highways.
It is easiest to understand the computation in the context of a specific example. Consider
model specification (1) for Highway A10. The expected number of tickets written per
car on an unannounced day is given by the probability that a car speeds (0.031 from the
first row, labeled (a)) times the probability that it is monitored (pL = 0.0084):
1 ∗ (.031) ∗ (.0084) = 2.604 × 10−4 .
On an announcement day, the probability that a car speeds is 6.5% lower (from the third
row), and the probability that it is monitored is equal to 20.07%. The expected number
of tickets written per car is therefore:
1 ∗ (.031)(1 − 0.065) ∗ (.2007) = 5.8203 × 10−3 .
Therefore, an increase in the budget of 10, 000 tickets allows subjecting to announcement
10, 000
= 1.7986 × 106
5.8203 × 10−3 −2.604 × 10−4
additional cars. Without an announcement, 3.1% would speed. Of these, 6.5% are
deterred from speeding by the announcement. Thus,
1.7986 × 106 ∗ 0.031 ∗ 0.065 = 3, 597
drivers will be deterred as a result of writing 10, 000 more tickets which will be issued
on additional announcement days. The same calculation is performed to obtain “Effect
of additional 10, 000 tickets” in Table 4. Depending on the model specification, the
reduction in the number of speeders ranges from 2,743 to 9,603 on highway A10, from
3,849 to 10,291 on highway A14, and from 763 to 3,817 on R4.
There is a large literature documenting the effect of speeding on accidents, injuries and
traffic deaths.42 We can use estimates from the literature of the impact of speeding on
42
For the US, the National Highway and Transportation Safety Authority (NHTSA) provides estimates
of speed-related crashes.
33
fatalities along with our estimates of how additional resources reduce speeding to evaluate
whether the police optimally trade-off the costs and benefits of speeding interdiction. We
take an average of the estimated deterrence effects of 10,000 tickets, found in Table 4 to
be about 4,000 speeders. Assuming that each deterred motorists travels the length of
a sector (about 40 km), 160,000 km are travelled by deterred motorists. The expected
number of deaths on 480,000 travelled kilometers is around
1.3
·160000
100000000
44
In our data, deterred motorists reduce their speed by about 8 km/h.
= 0.00208.43
Assuming the
45
probability of injury and death increases by 5% per km/hour, the additional interdiction
is expected to reduce the number of deaths by 40%, or by 0.00208 ∗ 0.4 = 8.32 × 10−4 .
On the cost side, writing 10,000 more tickets costs $5,000 in administrative costs
and, we assume, wastes about 1 minute per deterred driver, or a total of about 67 hours.
Given a wage of $10/h (the opportunity cost of time), the total costs of interdiction is
$5, 000 + 670.
If the police were resolving optimally the trade-off between marginal cost of interdiction and marginal benefits, in terms of statistical lives saved, then the implied value
of a statistical life is
5670
8.32×10−4
= 6.8 million dollars.
This value is within the range
of commonly used estimates of the value of a statistical life, indicating that the use of
resources in policing may be close to efficient.46
43
We impute the expected number of deaths at 1.3 per 100 million km travelled (See DiGuiseppi et al
(1998).) For comparison, in the US, where the speed limit is lower, the expected number of deaths was
1.51 per 100 million of highway miles travelled in 2002 (see Motor Vehicle Traffic Crash Fatality Counts
and Injury Estimates for 2003 ).
44
Average speeds among speeders are 142 and 144 respectively for A10 and A14. Assuming that
deterred motorists travel at the maximal non-ticketed speed (135km/h), deterred motorists reduce their
speed by 7 - 9km/hr.
45
This estimate is given in Finch et al. (1994). This estimate was used by the Belgian police in an
internal memorandum to evaluate the impact of speeding on casualties.
46
For example, Murphy and Topel (2003) report a range of $3 million to $7 million. The Environmental
Protection Agency in the U.S. uses an estimate of 4.5 million. (Murphy and Topel, 2003). If we do
the same calculation using an estimated effect of 12,000 vehicles deterred per 10,000 tickets, we get an
implied value of a statistical life of 2.86 million dollars.
34
4
Discussion and Some Extensions
This section compares our theory to some other existing theories of crackdowns. It also
extends the model to capture an environment in which crime is not a binary decision
but rather a continuous choice and the utility of the citizens is not necessarily linear
(to allow for risk aversion). The previously established theoretical results carry over to
these more general settings. Finally, we also consider some variants of the base model
and derive analogous results for these variants.
4.1
4.1.1
Alternative theories of deterrence
Rational Theories
As noted in the introduction, Lazear (2006) independently analyzes an auditing model
which is close to the one analyzed here, and so deserves careful discussion. The main
focus in Lazear’s paper is on academic testing as an incentive for pupils to study a subject.
The test can only include a given number of questions and so it only has a limited power
to incent. This power can be spread thinly across the entire subject matter or focused on
specific areas. Unlike us, Lazear (2006) does not investigate the optimal testing strategy.
Instead, that work focuses on the following specific strategy: the students are told that
a fraction (1 − q) of the subject matter will not be tested at all and that the test will
only cover the remaining q of the material. Note that q < 1 would correspond, in our
language, to a special type of crackdown with pL = 0.
An auditing policy of this type is, in general, suboptimal because it constrains pL to
be zero. The effects of this restriction can be seen in Figure 4 where for a given P the
optimal crackdown intensities (marked by pL and pH ) are compared to the constrained
crackdown intensities (indicated by 0 and lH ). As expected, under the optimal crackdown
35
the crime rate (S) is lower than under the restricted crackdown (Sl ).
Fraction
not
learning
Lazear’s
crackdown
Sl
S
1-F(pT)
0
pL
P
lH
pH
p
(testing
intensity)
Figure 4. Optimal versus Lazear’s Crackdown.
Observe that restricting the no-crackdown probability to zero also distorts the crackdown probability (lH ) relative to the optimal one. Indeed, it is easy to see that, regardless
of the shape of F , it must be lH ≤ pH . We may also note that in Lazear’s treatment,
crackdowns are inefficiently rare, in the sense that crackdowns are adopted for fewer
values of P than is optimal (again, this follows by inspection of Figure 3). For values
of P when crackdowns arise in Lazear’s framework, the estimated deterrence effect as
computed using Lazear’s crackdowns exceeds the deterrence effect in the optimal scheme
(in Figure 3, the slope of the “Lazear crackdown” is steeper than the slope of the optimal crackdown). Finally, under the optimal policing scheme the deterrence effect of
additional resources is subject to the law of diminishing returns: as P increases, its marginal effect on crime never increases (the deterrence effect is given by the slope of the
lower envelope of the function 1 − F (pT )). In Lazear’s treatment, the law of diminishing
returns does not hold: deterrence is generally non-monotonic in P .
There seem to be few practical reasons to restrict attention to a scheme where pL = 0.
The Belgian police, for example, spend considerable resources on non-crackdown interdiction. In the context of academic testing, the optimal scheme could readily be imple36
mented by telling students that a fraction (1 − q) of the subject matter is less likely to be
tested, but not necessarily out of bounds. One might then split the test into two parts,
each covering fractions q and (1 − q) of the subject matter, and devote an appropriate
number of questions to each part. The optimal auditing scheme is then implemented by
setting q = µH and letting pH determine what number of questions are allocated to the
high-intensity testing. Because lH ≤ pH , in Lazear’s scheme even those subjects that are
stressed for the test will be stressed insufficiently relative to the optimal scheme.
Despite the difference in approach, it is reassuring that in one respect the two analyses
give the same prediction: as the budget constraint is relaxed, both approaches imply that
the fraction q of the material subjected to high-intensity testing increases, and that the
per-unit-of-material intensity of testing remains unchanged (even though the latter is
suboptimal in Lazear, 2006).
4.1.2
Boundedly Rational Theories
The model analyzed in this paper is a model of perfectly rational risk assessment, in
which motorists form a Bayesian update of the probability of being monitored based on
the available information. In this environment, we have shown that crackdowns may be
part of the deterrence policy chosen by the police.
Alternatives to our theory exist that can rationalize crackdowns. These theories are
typically based on some form of non-standard (at least from an economist’s viewpoint)
rationality. For example, an alternative model of the effect of crackdowns is the following.
Suppose that absent a crackdown, the probability of monitoring is so small as to be
ignored by the driver. Crackdowns raise the speeder’s probability of detection to the point
where it is not negligible, and in the process the motorist becomes alert to the detection
risk which was previously unforeseen. If this increased alertness persists even when a
crackdown is not in force, crackdowns help reduce speeding. In our speeding application,
this theory of deterrence would suggest that motorists on announcement days would be
reminded of the possibility of being monitored and therefore slow down. According to
this theory, speeding should also go down even on roads that are not monitored due to
the increased alertness from announcements on other roads. This implication, however,
is refuted by our data, because we find that the fraction of speeders on non-monitored
37
roads does not decrease during monitoring days (see Table A4).
Criminologists have justified crackdowns using an alternative theory of deterrence,
based on subjective risk assessment. This theory, developed in Ross (1984) and Sherman (1990), highlights the distinction between risk (which is accurately perceived by
motorists) and uncertainty (which is not accurately perceived). The idea is that crackdowns, because they are fleeting, generate doubt in the mind of the motorist about the
interdiction intensity at any particular time, thus boosting the uncertainty component
involved in the decision to speed. According to this theory, using crackdowns may magnify the deterrence effect obtained from a given amount of resources.47 According to
this theory, an effective policing strategy would leave motorists in as much in doubt as
possible as to the location and timing of the crackdowns, in order to maximize their
uncertainty. This implication appears to contrast with the actual policy of liberal information dissemination observed in our application. Note that, consistent with the
observed pattern of information dissemination, the optimal policy in our model is to
inform motorists about the crackdowns.48
4.2
Crackdowns persist if citizens’ utility function is nonlinear
The model developed in this paper assumes that the utility function was linear in the
benefit, x, from committing a crime. The linearity assumption can be relaxed. Suppose
citizens have a utility function u that is increasing in x. Then they will commit a crime
iff
(1 − p) u (x) + pu (x − T ) > u (0) .
Consider the set of values of x such that the inequality is satisfied, and denote by H (p)
the measure of this set. The function H (p) represents the crime rate. Note that since u
is increasing, the left hand side is increasing in x and, also, u (x) > u (x − T ) whereby
47
Sherman (1990) goes further and argues that the beneficial effect of crackdowns on the motorists’
uncertainty about interdiction actually persists even after a crackdown period is over. He call this effect
“residual deterrence,” and argues that residual deterrence is an important component in the effectiveness
of the crackdowns.
48
Our model predicts the effectiveness of the crackdown is directly proportional to the fraction of
motorists who are aware of it. If no motorist were aware of the crackdown, all motorists would expect
the average amount of interdiction and there would be no effect of crackdowns on speeding.
38
the left hand side is decreasing in p. Therefore, the set of values of x such that the
inequality is satisfied decreases as p increases. This means that H (p) is decreasing in p.
The analysis of Sections 2 and 2.3 can then be carried out replacing F with H.
4.3
Crackdowns persist if the crime decision is continuous
Suppose that instead of a binary problem (committing a crime or not), each citizen solves
a more complicated problem involving not only whether to commit a crime, but also the
degree to which to commit it. For example, a motorist may choose whether to speed
and how much to speed. Suppose that the penalty for driving at s miles per hour above
the speed limit is an non-decreasing function T (s) (which could be equal to zero below
the speed limit) and that the agent’s utility from exceeding the speed limit by s is an
increasing function x (s). We allow different individuals to have different functions x (s).
Given a certain level of interdiction p, an agent with a given function x (·) solves
max x (s) − pT (s) .
s
Denote with s∗ (p) the maximizer of this problem. Denote by Fe (s|p) the fraction of
individuals who choose to travel at or below speed s for given p. The quantity Fe (s|p)
will depend on the distribution of the functions x (·) that is present in the population.
It is easy to see, however, that the optimal speed s∗ (p) is decreasing (or at least not
increasing) in p : s0 (p) = T 0 / (x00 − pT 00 ) is negative by the concavity of x − pT at the
maximum. This means that any motorist, regardless of his or her x (·) , will decrease
his or her optimal speed as the probability of being monitored increases. The function
Fe (s|p) is therefore increasing in p.
If police cared not only about the fraction of people who exceed the speed limit, but
also about their speeding levels, the police’s objective function would be represented by
the function
D (p) ≡
Z
K (s) dFe (s|p) ,
where K (s) is some non-decreasing function. The function K (s) represents the disutility
that the police receives from having one motorist travel at speed s. Because Fe (s|p) is
increasing in p, the function D (p) is decreasing in p. Now, rewrite problem (2) replacing
1 − F (p) with D (p). This yields a mathematical formulation of the problem in which
39
motorists can choose the amount of speeding and police care not only about the fraction
of speeders but also about their speed. From a formal viewpoint this new formulation
is similar to the original problem. Therefore, all the qualitative features of the solution
to the original problem carry over, including the optimality of crackdowns when the
function D (p) exhibits non-convexities.
4.4
Size of the penalty
In considering the agency relationship between the principal and the police we have taken
the size of the penalty (T in the formal model) to be exogenous. In actuality, the size
of the penalty is an object of choice, sometimes on the part of the principal.49 It should
be pointed out that our theory remains meaningful irrespective of how or by whom the
penalty is chosen, as long as deterrence is not perfect. Thus, the theory does not require
us to explain how the penalty is determined.
One might ask, however, why would deterrence not be perfect. Or, a more meaningful
question in our applied setting, why would penalties not always be set at their maximal
value so that everyone is deterred with a minimum deterrence cost? In our application,
for example, we might ask why speeders are not sent to jail? This point was raised by
Becker (1968) who noted that, when interdiction is costly, increasing the size of the fine
allows the same level of deterrence to be implemented while saving on interdiction costs.
Regardless of the objective to be implemented, then, penalties should always be set at
their maximal value.
Why are observed penalties not always maximal? The literature has identified several
arguments; here we mention those that are more directly applicable to our analysis, and
refer to Polinsky and Shavell (2000) for an excellent overview of the others. First is
the need to generate marginal deterrence: if all violations were punished with the same
(maximal) intensity, there would be no incentive to choose a lesser over a larger (and more
harmful) violation (see Stigler 1970, and more recently Shavell, 1992). If enforcement
cannot be tailored perfectly to the magnitude of the violation, lower penalties will need
to be applied to less serious crimes. A second reason is that, when a harsh penalty is
imposed, those who enforce the law must be monitored lest they abuse their position,
49
In our speeding application the fines are chosen by the Belgian parliament.
40
and opportunities for judicial appeal must be provided to redress enforcement mistakes.50
Taking into account these ancillary (but very important) costs may explain why penalties
are seldom set at their maximal possible value. A third consideration is risk aversion
(see Polinsky and Shavell 1979): whenever it is socially optimal for some individuals to
violate the law (for example, speeding may be socially optimal in some circumstances),
increasing the penalty and decreasing the risk of apprehension increases the risk faced
by these “optimal violators,” and thus reduces social welfare. This consideration places
limits on the size of the penalty chosen by the social planner. While we do not explicitly
model these considerations, they could easily be added to the model without affecting
the fundamental force that generates crackdowns. For example, if the principal cannot
observe the crime rate, the police must be given incentives based on some other measure
of performance.
4.5
Other Applications
The ideas developed in this paper have potential applications beyond speeding. In this
section, we consider some of them.
4.5.1
Tax Evasion and Drug Interdiction
Suppose the principal cares about reducing the crime of drug production, but the principal can only observe the drugs that make it to the market without being intercepted.
In that case, police performance will have to be evaluated based on undetected crime.
Sometimes, minimization of undetected crime may even arise as a first-best option. For
example, a principal may find it optimal to give incentives to the police based on undetected crime when detection removes the social cost of the crime. This is the case for
example with illegal firearms, where if a firearm is intercepted it is taken off the street.
The same is true for tax evasion. The objective of the tax authority is to minimize
undetected crime.
When a group is monitored with intensity p, the fraction of crime in the group that
goes undetected is (1 − p) (1 − F (pT )). Given a policing strategy µ, undetected crime
50
So, for example, the degree of judicial protection is much larger (and the system therefore much
more expensive) in death penalty cases than for traffic violations.
41
is given by
Z
0
p
µ (p) (1 − p) (1 − F (pT )) dp.
(7)
The police chooses a policing strategy µ to minimize expression (7) subject to the budget
constraint (3).
This programming problem is very similar to the one studied in Section 2.1; there as
well as here, the objective function is decreasing in p. This was the only property of the
objective function that was used in Section 2.1, so it is immediate that Propositions 1
and 2 continue to hold in this setting.
Whether crackdowns are optimal depends, as before, on the convexity of the objective
function. In the present case, it is the convexity of undetected crime that matters. If
undetected crime is convex in p then crackdowns are never optimal (see Remark 1.) It is
simple to verify that undetected crime is “more convex” than crime, in the sense that if
(1 − F (pT )) is convex then (1 − p) (1 − F (pT )) is also convex. Therefore, if F is such
that police minimizing overall crime rates never finds it optimal to engage in crackdowns,
then crackdowns are also not optimal if the objective is to minimize undetected crime.
4.5.2
Internal Compliance
Most large corporations employ internal compliance officers who monitor and audit the
processes of the corporation’s internal departments. Much of this monitoring is directed
at ensuring that the firms is in compliance with legal regulations. Some of this monitoring
takes place on a continuous basis, such as oversight of contracts, of insurance policies,
etc. But some of this monitoring is designed to be carried out episodically, such as when
a review is made of a particular department within a company. The combined effect of
these monitoring activities is arguably to create a crackdown policy, where a mild level
of monitoring is sometimes augmented during a crackdown.
There are arguably two related reasons for the crackdowns. One is that the decision
makers in the department are only deterred by a very high monitoring intensity, higher
than can be provided on a continuous basis. Then, just as in the model of Section 2.1,
crackdowns can help maximize compliance. A second reason why crackdowns are employed is a technological nonlinearity in the scope of the monitoring. When departments’
42
activities are audited, it is on average easier to evaluate compliance if the activity of a
few is monitored thoroughly, rather monitoring all but in little depth. We can imagine,
then, that the probability of detection is nonlinear (and convex) in the scope of the
monitoring. At the extreme, it might be impossible to detect wrongdoing unless all of
a given department’s activity are scrutinized. This technological nonlinearity is related,
though not exactly identical, to the model studied in Section 2.1.
4.5.3
Collecting TV-license Fees in Sweden
Under Swedish law everyone who owns a television set is required to pay the licence fee.
The fee is collected by Radiotjänst, a private corporation that administers the TV-fee
as well as checks out that people are actually paying the fee. The controls are carried
out throughout the year. In addition, there are special fee-controls. These are stricter
controls in 3-5 predetermined areas every year. According to the law on the TV license
fee, everyone should be informed of their area being subjected to special fee-control.
Among other things, this is done by postcards being sent out to all households in the
area. Information is also given in advance through the media and radio, and through the
TV-detecting films that are shown on TV. The special fee-controls can be interpreted as
crackdown.
Consumers differ according to their taste for TV net of the cost of purchasing it,
captured by x, and their psychological cost of being fined, captured by h. The two parameters are realizations from a joint probability distribution (so we allow for correlation
between X and H). For all consumers, the scalar d represents the monetary cost of the
fee, and the value of not having a TV is zero.
In our setup, the socially optimal outcome is for all consumers with x > 0 to receive
the TV. However, consumers with x < min {ph, d} will opt not to get a TV. Thus,
interdiction (increasing p) entails inefficiencies due to exclusion of some consumers. On
the other hand, interdiction reduces cheating and thus increases the receipts from the
fee. We want the regulated firm who collects the fine to set interdiction so as to balance
fee receipts with the inefficiencies generated by interdiction.
Given an interdiction level p, the fraction of consumers who actually pay the fee are
those who prefer paying the fee to not having a TV at all, and who are also more afraid
43
to cheat than they dislike paying the fine. The fraction
ϕ = Pr (X > d)
prefers paying the fee to not having a TV at all. So the fraction of fee-paying consumers
is
ϕ · Pr (pH > d|X > d)
Let us now turn to the social cost of interdiction. While we can be sure that a fraction
ϕ of consumers will get a TV–if necessary, by paying the fee–the remaining (1 − ϕ)
may actually choose not to get a TV. Those consumers will chose not to get a TV when
their payoff from cheating is smaller than zero. So the fraction of consumers who do not
get a TV is given by
(1 − ϕ) · Pr (X − pH < 0|X < d) .
This fraction is a measure of the social loss due to interdiction (which is increasing in p).
Denote
FH (y) = Pr (H ≤ y |X > d)
µ
¶
X
FX/H (y) = Pr
≤ y|X < d .
H
Then the objective function is
µ
µ ¶¶
¸
Z p̄ ·
d
d · ϕ · 1 − FH
− α · (1 − ϕ) · FH/X (p) µ (p) dp.
max
µ
p
0
The first addend in brackets accounts for the receipts from the fee (increasing in p). The
(−α) coefficient represents the weight assigned to the social loss. If the term in bracket
is convex as a function of p then we have the possibility that crackdowns, such as are
observed in Sweden, are optimal.
5
Conclusions
This paper introduced and analyzed a model of police interdiction. Within this model,
we have fully characterized the optimal interdiction strategy. We have shown that even
if all citizens look identical to the police, it may be rational for the police to divide
44
the population into two (but no more) groups and monitor the groups at different intensities. For this division to be effective in curtailing crime, it is important that the
group subjected to the crackdown be made aware when they are being monitored at
the higher rate. This explains why police would announce when and where crackdowns
will occur. Our analysis provided a rational choice explanation for pre-announced police
crackdowns, which are regarded in the criminology literature as exploiting a non-rational
perception of risk on the part of the citizens.
The model provides a behavioral setting in which monitoring intensities could differ
across locations or groups in a way that is totally random, thus potentially providing a
rigorous basis for using the observed variations in monitoring intensities in an observational data to estimate the deterrence effect without having to worry about endogeneity.
We applied our theoretical model to study speeding interdiction in Belgium. The
data provide support for several implications of the model. Among these are, first, that
the announcement strategy of the police indeed amounts to dividing the population into
exactly two groups, and, second, that when police resources are increased, the frequency
of crackdowns increases but the probability of being policed during a crackdown does
not change. We used the model to estimate the deterrence effect of additional resources
devoted to speeding interdiction in the form of 10,000 additional speeding tickets. Our
calculations suggest that the marginal benefit, in terms of statistical lives saved, is close
to the marginal cost of deterrence (it is exactly equal if we take the value of a statistical
life to be 6.85 million dollars). Thus, the current level of speeding interdiction is arguably
in line with socially optimal use of resources.
The standard theoretical approach we adopted–set up a stylized model and derive
the optimal policy–turns out to be remarkably successful in explaining the policing
behavior of a police department. What is remarkable is that the optimal interdiction
strategy in our model has some features (crackdowns) which are unlikely to be put in
place by chance, and which are found in the Belgian data. Thus, the rational theory of
optimal interdiction can have not only normative implications, but even positive correlates in practical police work. If apparently abstract models can capture the behavior of
complex real-world institutions such as police departments, then this is good news for
the vast theoretical literature dealing with optimal enforcement.
45
Table 1a
Number and Percentage of Vehicles Subject to
Announced or Unannounced Monitoring by Year§
2000
2001
2002
2003 (first
half of year)
Announced
266,240
394,540
1,746,340
1,777,977
(39.5%)
(55.2%)
(76.8%)
(58.0%)
Unannounced
406,941
319,650
526,422
1,139,428
(60.5%)
(44.8%)
(23.2%)
(42.0%)
Total
673,181
714,190
2,272,762
2,917,405
§Based on daily observations on the number of vehicles, as recorded by the radar control machines.
Table 1b
Number of Ticketed Speeders by Year and Announced or Unannounced Monitoring
(percentage of total ticketed each year shown in parentheses)§
2000
2001
2002
2003 (first
half of year)
Announced
Number ticketed
12677
20039
55804
32079
(37.3%)
(44.3%)
(71.4%)
(69.4%)
Unannounced
Number ticketed
21274
25225
22332
14140
(62.7%)
(55.7%)
(28.6%)
(30.6%)
§ Based on daily observations on vehicles passing by the radar control machines and on their speeding
status, as recorded by the machine.
A10
A14
R4
Ann
18
38
10
Total
66
µ h†
Table 1c
Number of Announced (Ann) and Total Monitoring Events
by Highway and Year
2000
2001
2002
Total
Ann
Total
Total
Total
46
23
52
181
214
138
51
105
156
244
34
1
24
0
5
218
0.012
75
181
0.014
337
463
0.062
2003 (first half of year)
Ann
Total
125
158
150
218
0
0
275
376
0.100
† µh is calculated as the number of announced monitoring events divided by the total number of potential
monitoring events (5475 in 2000-2002 and 2715 in the first half of year 2003), which includes intervals
when monitoring took place as well as intervals in which no monitoring took place). 5475 is calculated as
number of road segments (5) times the number of intervals per day (3) times the number of days per year
(365).
Table 2
Means of Variables†
Variable
Fraction speeding
Fraction vehicles traveling on highway A10
Fraction vehicles traveling on highway A14
Fraction vehicles traveling on highway R4
Quarter 1 (Jan, Feb, Mar)
Quarter 2 (Apr, May, Jun)
Quarter 3 (Jul, Aug, Sep)
Quarter 4 (Oct, Nov, Dec)
Year 2000
Year 2001
Year 2002
Year 2003
Heavy traffic density
Medium traffic density
Moderate traffic density
Light traffic density
Weekday morning rush hour
Weekday evening rush hour
Holiday
Mean
(Std. err)
0.04
(0.004)
0.43
(0.010)
0.56
(0.01)
0.02
(0.003)
0.31
(0.009)
0.34
(0.010)
0.15
(0.007)
0.21
(0.008)
0.11
(0.006)
0.11
(0.006)
0.40
(0.01)
0.37
(0.01)
0.02
(0.003)
0.001
(0.0007)
0.006
(0.002)
0.97
(0.003)
0.02
(0.003)
0.29
(0.009)
0.08
(0.005)
† Observations are monitoring intervals. Means are weighted by the number of
vehicles in each interval. There are 1238 total monitoring intervals (see Table 1c
for breakdown by year)
Table 3
Estimated Coefficients from Logistic Regression of Probability of Speeding§
Dep. Var: Indicator for Whether Speeding
(Standard errors shown in parentheses)
Model Specification
Variable
(1)
(2)
(3)
Intercept
Announcement on highway A10
-3.43
(0.007)
0.58
(0.009)
0.35
(0.02)
…
-3.46
(0.02)
0.27
(0.008)
-0.16
(0.02)
…
Announcement on highway A14
…
…
Announcement on highway R4
…
…
Medium traffic density†
-0.93
(0.04)
-0.96
(0.02)
-0.38
(0.15)
…
Moderate traffic density†
…
Light traffic density†
…
Indicator for weekday morning rush hour
…
Indicator for weekday evening rush hour
…
Indicator for holiday
…
Includes fixed effects for days of week
Includes fixed effects for months of year
Includes fixed effects for year
No
No
No
-0.28
(0.04)
-0.39
(0.02)
-0.08
(0.15)
-0.28
(0.06)
0.24
(0.03)
0.22
(0.01)
-0.45
(0.02)
0.02
(0.006)
0.51
(0.007)
Yes
Yes
Yes
Indicator for highway A14
Indicator for highway R4
Probability of monitoring – A10‡
Probability of monitoring – A14‡
Probability of monitoring – R4‡
-3.41
(0.02)
0.19
(0.008)
-0.23
(0.02)
-0.18
(0.009)
-0.11
(0.006)
-0.04
(0.05)
…
…
…
-0.25
(0.06)
0.27
(0.03)
0.24
(0.01)
-0.44
(0.02)
-0.02
(0.006)
0.49
(0.007)
Yes
Yes
Yes
Number of vehicle observations
5,641,522
5,641,522
5,641,522
% correctly classified under the model
44.0%
48.02%
48.3%
§ The unit of observation is a vehicle passing by the radar control machine. The speeding
status is recorded by the machine.
† The omitted category is high traffic density.
‡ The probability of monitoring was obtained using the procedure described in the text, which takes
into account the expected probability of monitoring, the expected length of time spent monitoring,
and uncertainty about the direction of the road being monitored.
Table 4
Decrease in Speeding Attributable to a Crackdown and the Deterrence Effect
of Increasing the Number of Tickets
Avg monitoring
Model Specification
probability§
pH=0.2007
pL=0.0084
pH=0.2441
pL=0.0167
pH=0.2576
pL=0.0091
Highway A10
Predicted % speeding on days without
announcements (a)
Decrease in % speeding due to the
announcement (b) †
Percentage decrease in rate of speeding *
Implied slope of 1-F ¶
Effect of additional 10,000 tickets on number of
speeders ₤
Highway A14
Predicted % speeding on days without
announcements (a)
Decrease in % speeding due to the
announcement (b) †
Percentage decrease in rate of speeding *
Implied slope of 1-F ¶
Effect of additional 10,000 tickets on number of
speeders ₤
Highway R4
Predicted % speeding on days without
announcements (a)
Decrease in % speeding due to the
announcement (b) †
Percentage decrease in rate of speeding *
Implied slope of 1-F ¶
Effect of additional 10,000 tickets on number of
speeders ₤
(1)
3.1
(2)
3.0
(3)
3.1
0.20
0.15
0.48
6.5%
-1.04%
-3,597
5.0%
-0.78%
-2,743
15.5%
-2.50%
-9,603
5.4
5.0
5.0
1.01
0.40
0.48
18.7%
-4.44%
-10,291
8.0%
-1.76%
-3,849
9.6%
-2.11%
-4,707
4.4
4.3
4.2
0.38
0.08
0.16
8.6%
-1.53%
-3,817
1.9%
-0.32%
-763
3.8%
-0.64%
-1,596
§ The monitoring probabilities pH and pL are derived following the procedure described in the text, which
takes into account the probability of monitoring, the expected length of time spent monitoring, and
uncertainty about the direction of the road being monitored. pH is the average probability of being
monitored during an announcement period and pL is the probability of being monitored when there was no
announcement.
† Change in speeding rate implied by the estimated coefficients from the logistic regression model (Table
3) when the monitoring probability increases from pL to pH.
* (b)/(a)*100
¶ Slope = (b)/(pL - pH)
₤ As described in the text.
Appendix A: Proofs
Theorem 1 Let the function f : [0, S] → [0, 1] be continuous and strictly increasing.
Let the function g : [0, S] → R be continuous. Let Pg∗ denote the set of probability
distributions defined on the interval [0, S] that solve the following linear problem
Z S
f (p) µ (p) dp
max
µ
0
Z S
g (p) µ (p) dp ≤ C.
s.t.
(8)
0
For given f, let Fg denote the set of all functions f with the property that all µ∗ ∈ Pg∗
place all the probability on one or two points in [0, S]. Then, the set Fg is dense in the
set of all continuous functions g equipped with the supnorm.
If, moreover, the solution requires that probability mass be placed on two points in
[0, S], then the same two points receive all the probability when C is slightly increased.
Proof: Consider first the easy case in which the constraint is not binding at the
optimal solution. In that case, a generic f will have exactly one strict maximum, and so
the optimal µ∗ will put mass one on exactly one point (the strict maximum).
Let us now consider the more difficult case in which the constraint is binding at the
optimal solution. In that case, there exists a number λ > 0 such that µ∗ maximizes the
Lagrangean
L(µ, λ) =
Z
0
S
[f (p) − λg (p)] µ (p) dp + λC.
We will show that, if µ∗ ∈ Pg∗ puts positive mass on more than two points, then f is
non-generic. To this end, let A denote the set of p’s that is defined by
A = arg max [f (p) − λg (p)] .
p
By definition of A, there is a number M such that
f (p) − λg (p) = M
f (p) − λg (p) < M
for p ∈ A
for p ∈
/A
If µ∗ puts positive mass on more than two points, then the cardinality of A would have to
exceed 2. Consider the transformation ϕ (p) = f −1 (p/S). The function ϕ is a one-to-one
mapping of [0, S] onto itself. We can therefore write
f (ϕ (p)) − λg (ϕ (p)) = M
f (ϕ (p)) − λg (ϕ (p)) < M
for ϕ (p) ∈ A
for ϕ (p) ∈
/ A,
or, with the obvious meaning of symbols,
p
S
p
S
− λg (ϕ (p)) = M
− λg (ϕ (p)) < M
for p ∈ ϕ−1 (A)
for p ∈
/ ϕ−1 (A) .
Note that the set ϕ−1 (A) has the same cardinality of A. Thus, if A has cardinality
greater than 2, it means that the two numbers λ and M are such that the negatively¢
¡
sloped straight line identified by λ1 Sp − M is tangent to the function g (f −1 (p/S))
at more than two points and never exceeds it. This means that there is a tangent
hyperplanes to the set
©
¡
¢ª
Y = (p, y) : y ≤ g f −1 (p/S)
which makes contact with the set Y at more than two points and has negative slope.
We now show that, the set of f ’s such that this property does not hold is dense. To
this end, and without loss of generality, let us assume that S = 1. Our task, then, is
to show that if a negatively-sloped tangent hyperplanes to Y make contact with Y in
more than two points, there is a function f˜ close to f with the property that no tangent
hyperplane has more than two contact points. Let H denote the set of hyperplanes that
have more than two contact points with Y . Elements of H are identified by their slope
h. For every hyperplane h ∈ H, take the sup and the inf of the first dimension of all its
contact points and call those ah and bh . Consider now a continuous function dg (p) which
is equal to 0 for every p unless p ∈ (ah , bh ) for some h ∈ H, in which case dg (p) assumes
˜
values strictly
n between zero and
o 1. Let fε (p) ≡ [1 + ε · dg (p)] · f (p). For any ε > 0,
the set Ỹε = (p, y) : y ≤ f˜ε (p) has exactly the same set of tangent hyperplanes as Y .
This follows from the fact that since the functions f and g are continuous, hyperplane h
makes contact with Y at ah and bh . Moreover, by construction no hyperplane is tangent
to Ỹε at more than two points. Since the function f˜ε (p) can be made arbitrarily close to
f (p) in the supnorm by choosing ε to be small, the set Fg is dense.
Let us now turn to the second part of the statement. For given C, suppose that the
solution requires placing probability mass on two points pL < pH . Then, the constraint
must be binding. To see this, define
pm ≡ arg
pM ≡ arg
min
p∈{pH ,pL }
f (p)
max f (p) .
p∈{pH ,pL }
Since f is strictly monotone, f (pM ) > f (pm ), and the only reason why it is optimal to
place any probability mass on pm is to help satisfy the constraint. It must therefore be
g (pM ) > C > g (pm ) . At the optimal solution, moreover, it cannot be optimal to place
anything but the smallest probability mass on pm so that the constraint is just satisfied
(with equality). Denote by λ∗ the Lagrange multiplier associated to this programming
problem. Since f is strictly monotone, λ∗ > 0.
e = C+ε
Suppose now that the constraint is relaxed slightly, by increasing C to C
with ε³a small
´ positive number. The solution to the programming problem is a saddle
e for the Lagrangean. We now proceed to construct this saddle point. We
point µ
e, λ
e = λ∗ . Because of this choice, the
start by keeping the Lagrange multiplier unchanged, λ
µ
e that maximizes the Lagrangean still places probability mass on pM and pm only, which
is what we wanted to prove. To conclude the proof we need to finish the construction
e = λ > 0 to minimize the
of the saddle point. To this end, observe that in order for λ
Lagrangean, the Lagrangean must be constant with respect to λ, which is equivalent to
e
e (pm ) + g (pM ) µ
e (pM ) = C
g (pm ) µ
(9)
e > g (pm ). Therefore,
Since g (pM ) > C > g (pm ), for ε sufficiently small also g (pM ) > C
it is possible to choose µ
e (pm ) and µ
e (pM ) = 1 − µ
e (pm ) so that equation (9) is satisfied.
Choosing µ
e accordingly concludes the proof.
Corollary 2 If f is increasing and the solution requires that probability mass be placed
on two points in [0, S], the probability mass placed on the largest point increases when C
is slightly increased.
Proof. >From the proof of Theorem 1 we know that the constraint (8) is binding both
at C and at C + ε. This means that for c = C, C + ε , the probability mass µc placed on
pH must solve
g (pH ) µc + g (pL ) (1 − µc ) = c.
Since f is increasing, pM = pH and thus g (pH ) > g (pL ) . The result follows.
Appendix B: Additional Tables
Table B1
Number of Announced and Unannounced Monitoring
Events by Month and Year on all Highways†
2000
2001
2002
2003 (first half of
year)
Ann
Unann
Ann
Unann
Ann
Unann
Ann
Unann
January
February
March
April
May
June
July
August
September
October
November
December
3
10
5
3
3
4
7
8
4
3
3
13
7
12
6
18
10
14
13
7
9
18
17
21
10
7
5
6
6
4
6
7
6
5
6
7
12
9
3
4
8
11
10
8
7
5
13
16
15
17
19
24
29
27
31
32
29
41
36
37
17
9
26
21
11
8
6
1
6
12
6
3
41
45
32
36
64
57
*
*
*
*
*
*
3
9
24
24
21
20
*
*
*
*
*
*
Total
66
152
75
106
337
126
275
101
†Sample includes all monitoring events.
Table B2
Number of Announced and Unannounced Monitoring
Events by Day of Week and Year on all Highways†
2000
2001
2002
2003 (first half of
year)
Ann
Unann
Ann
Unann
Ann
Unann
Ann
Unann
Saturday
Sunday
Monday
Tuesday
Wednesday
Thursday
Friday
8
17
8
7
10
6
10
38
29
17
15
16
19
18
23
10
6
12
13
7
4
27
34
11
8
9
9
8
40
64
24
68
58
39
44
19
31
12
17
14
22
11
36
29
48
40
37
43
42
6
20
11
17
20
11
16
Total
66
152
75
106
337
126
275
101
†Sample includes all monitoring events.
Table B3
Estimated Logistic Model for the Probability of Monitoring
when there is no announcement, by Year and by Road†
Dep. Var: Indicator for Whether Monitoring Occurred
Highway
Variables*
A10
A14
R4
Intercept
-2.50
-2.43
-4.36
(0.36)
(0.36)
(0.58)
quarter 1
0.37
-0.12
0.45
(0.25)
(0.44)
(0.45)
quarter 2
-0.19
0.44
0.55
(0.28)
(0.20)
(0.43)
quarter 3
-1.90
-0.47
0.11
(0.49)
(0.22)
(0.44)
announced last week
0.15
-0.12
0.04
(0.35)
(0.20)
(0.49)
announced yesterday
-0.22
-0.29
…
(0.41)
(0.23)
monitored last week
0.78
1.95
2.11
(0.40)
(0.36)
(0.34)
monitored yesterday
0.37
0.48
-1.21
(0.36)
(0.19)
(0.64)
some announcement same
0.71
-0.84
-1.34
day on any road
(0.23)
(0.20)
(0.50)
year 2001
-0.15
-0.67
0.28
(0.29)
(0.20)
(0.34)
year 2002
-0.10
0.27
-0.26
(0.32)
(0.20)
(0.54)
year 2003
-0.01
0.41
…
(0.31)
(0.24)
Number of observations
1273
1302
1451
% Correctly classified
76.1%
73.4%
80.7%
† The sample consists of three potential monitoring events for each day of the year,
excluding those events during which an announcement was made. The covariates are used
to predict whether there was monitoring during events when there was no announcement.
* The specification also includes fixed effects for day of week.
Table B4
Expected Length of Time Spent Monitoring, by Year and By Road
Dep Var: Hours Spent Monitoring During a Given Event†
Highway
A10
A14
Ann
Unann
Ann
Unann
Intercept
0.56
0.16
0.25
0.14
(0.22)
(0.08)
(0.24)
(0.07)
quarter 1
0.07
-0.03
0.01
-0.01
(0.06)
(0.05)
(0.06)
(0.04)
quarter 2
-0.06
-0.06
-0.10
-0.01
(0.05)
(0.05)
(0.06)
(0.03)
quarter 3
-0.06
-0.06
-0.01
0.06
(0.06)
(0.09)
(0.06)
(0.04)
year 2001
0.12
0.04
0.12
0.03
(0.11)
(0.05)
(0.08)
(0.03)
year 2002
-0.09
0.07
-0.02
0.07
(0.10)
(0.06)
(0.07)
(0.03)
year 2003
-0.13
0.14
-0.04
0.10
(0.10)
(0.06)
(0.08)
(0.04)
announced last week
-0.11
-0.0002
-0.04
-0.004
(0.11)
(0.06)
(0.07)
(0.03)
announced yesterday
-0.004
0.01
0.03
0.01
(0.07)
(0.06)
(0.06)
(0.03)
monitored last week
-0.01
0.002
0.22
-0.03
(0.06)
(0.06)
(0.11)
(0.05)
monitored yesterday
-0.004
-0.04
-0.04
0.01
(0.07)
(0.06)
(0.05)
(0.03)
some announcement same
…
-0.03
…
-0.05
day on any road
(0.03)
(0.03)
Medium traffic density
…
…
…
0.02
(0.15)
Moderate traffic density
-0.10
-0.06
…
0.07
(0.18)
(0.15)
(0.36)
Light traffic density
Holiday
Number of observations
R-squared
R4
Ann
0.40
(0.19)
0.29
(0.23)
0.05
(0.20)
0.34
(0.18)
…
…
Unann
0.32
(0.08)
-0.04
(0.04)
-0.09
(0.04)
0.001
(0.04)
0.002
(0.04)
0.02
(0.06)
…
-0.15
(0.17)
…
0.04
(0.04)
…
-0.03
(0.16)
…
0.01
(0.03)
-0.007
(0.05)
-0.04
(0.05)
-0.24
(0.08)
-0.21
(0.06)
…
…
-0.02
(0.23)
0.03
(0.13)
-0.19
(0.18)
0.06
(0.07)
346
0.002
(0.06)
0.02
(0.05)
122
0.08
(0.21)
0.02
(0.07)
394
0.11
(0.06)
-0.03
(0.03)
309
…
10
-0.19
(0.06)
0.03
(0.04)
51
0.12
0.11
0.07
0.07
0.84
0.45
…
† This regression is used to predict the length of time spent monitoring. The sample includes all
monitoring event, with and without announcements.
* All specifications also include fixed effects for day of week. The variable “holiday” was
not included in the above specifications because of too few observations.
Year
2000
highway
Table B5
Average Predicted Probability of Monitoring
no-announcement this
no-announcement
sector, announced other
sector
A10
A14
R4
2001
A10
A14
R4
2002
A10
A14
R4
2003
A10
A14
R4
* Too few observations in the cell.
0.004
0.011
0.009
0.004
0.011
0.014
0.008
0.027
0.001
0.011
0.030
*
0.009
0.003
0.002
0.007
0.003
*
0.010
0.007
0.001
0.018
0.020
*
Announcement
0.27
0.26
0.29
0.30
0.30
0.35
0.20
0.24
*
0.19
0.23
*
Table B6
Estimated Logistic Model for the Probability of Announcement by Highway†
Standard errors shown in parentheses
Dep. Var: Indicator for Whether Announced During Given Event
Highway
Variables*
A10
A14
R4
Intercept
-3.76
-4.07
-4.94
-4.90
-2.65
(0.36)
(0.61)
(0.40)
(0.41)
(0.82)
quarter 1
0.08
0.10
-0.004
0.02
-1.18
(0.21)
(0.22)
(0.20)
(0.21)
(1.18)
quarter 2
0.22
0.17
0.02
0.03
-0.20
(0.21)
(0.22)
(0.20)
(0.19)
(0.89)
quarter 3
-0.23
-0.23
-0.01
-0.003
0.48
(0.22)
(0.22)
(0.20)
(0.20)
(0.84)
Holiday
2.23
2.13
0.98
0.99
2.50
(0.47)
(0.50)
(0.27)
(0.27)
(1.34)
announced last week
0.34
0.26
-0.18
-0.17
-0.06
(0.36)
(0.38)
(0.21)
(0.22)
(0.96)
announced yesterday
-0.52
-0.52
0.19
0.19
…
(0.29)
(0.29)
(0.21)
(0.21)
monitored last week
1.38
1.41
2.65
2.63
-0.35
(0.41)
(0.41)
(0.38)
(0.39)
(0.80)
monitored yesterday
0.25
0.24
-0.31
-0.31
…
(0.27)
(0.27)
(0.19)
(0.19)
year 2001
0.06
-0.008
0.42
0.45
-2.39
(0.03)
(0.36)
(0.23)
(0.24)
(1.07)
year 2002**
2.13
2.31
1.55
1.53
…
(0.30)
(0.42)
(0.22)
(0.22)
year 2003
2.12
2.37
2.16
2.15
…
(0.30)
(0.49)
(0.24)
(0.25)
Fraction speeding if
…
0.05
…
-0.01
…
unannounced monitoring
(0.07)
(0.02)
Number of observations
1620
1620
1697
1697
732
-3.20
(1.46)
-1.19
(1.18)
-0.91
(1.82)
-0.06
(146)
2.28
(1.45)
0.48
(1.56)
…
-0.32
(0.80)
…
-2.67
(1.24)
…
…
0.14
(0.31)
732
p-value from test of joint
significance of all
<0.0001 <0.0001 <0.0001
<0.0001
0.9015
0.9116
covariates, except year
indicators
† The sample includes all monitoring events.
*All specifications include fixed effects for days of week. Some days of week indicators are
significant for A10 and A14 in 2000 and for A14 in 2001.
**There is only one announcement day on R4 in 2001 and none thereafter, so the R4 logistic
regression only includes years 2000 and 2001.
Table B7
Correlation of Announced Monitoring with Number of Speeders
and with the Proportion of Speeders on the Road
p-value shown in parentheses
Over all roads
Road A10
Road A14
Road R4
Number of Speeders
Proportion of Cars Speeding
-0.03
(0.31)
-0.21
(<0.0001)
0.05
(0.20)
-0.05
(0.70)
-0.26
(<0.0001)
-0.32
(<0.0001)
-0.77
(<0.0001)
-0.13
(0.33)
0.15
0.20
0.25
0.30
0.35
0.40
60
0.1
0.2
0.3
0.4
0.0
0.1
0.2
0.3
Ann Other Road, 2000
Ann Other Road, 2001
Ann Other Road, 2002
0.04
0.06
0.08
0.10
6
4
0
2
Frequency
6
4
0
2
Frequency
6
4
2
0
0.02
0.00
0.02
0.04
0.06
0.08
0.10
0.00
0.02
0.04
0.06
0.08
mean: 0.007
Unann Any Road, 2000
Unann Any Road, 2001
Unann Any Road, 2002
0.06
mean: 0.011
0.08
0.10
20
0
10
Frequency
20
0
10
Frequency
20
10
0.04
0.10
30
mean: 0.0036
30
mean: 0.0033
0.02
0.4
8 10
mean: 0.2432
8 10
mean: 0.3006
8 10
mean: 0.2622
0
0.00
40
0
0.0
30
0.00
20
Frequency
40
0
20
Frequency
40
20
0
Frequency
Frequency
0.10
Frequency
Ann. Control, 2002
60
Ann. Control, 2001
60
Ann. Control, 2000
0.00
0.02
0.04
0.06
0.08
0.10
mean: 0.0092
Figure 3: Probability of being monitored, highway A14
0.00
0.02
0.04
0.06
mean: 0.0243
0.08
0.10
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